Math Handbook of Formulas, Processes and Tricks Trigonometry

Math Ha d ook of For ulas, Pro esses a d Tri ks www. athguy.us Trigo o etry Prepared y: Earl L. Whit ey, FSA, MAAA Versio April Cop ight ‐ . , 7 7, Ea l Whit e , Re o NV. All Rights Rese ed Note to Stude ts This T igo o et Ha d ook as de eloped p i a il th ough o k ith a u e of High “ hool a d College T igo o et lasses. I additio , a u e of o e ad a ed topi s ha e ee added to the ha d ook to het the stude t’s appetite fo highe le el stud . O e of the ai easo s h I ote this ha d ook as to e ou age the stude t to o de ; to ask hat a out … o hat if … . I fi d that stude ts a e so us toda that the do ’t ha e the ti e, o do ’t take the ti e, to seek out the eaut a d ajest that e ists i Mathe ati s. A d, it is the e, just elo the su fa e. “o e u ious a d go fi d it. The a s e s to         ost of the uestio s elo a e i side this ha d ook, ut a e seldo taught. Is the e a ethod I a lea that ill help e e all the ke poi ts o a u it i le ithout e o izi g the u it i le? What’s the fastest a to g aph a T ig fu tio ? Ca I o e t the su of t o t ig fu tio s to a p odu t of t ig fu tio s? Ho a out the othe a a ou d, ha gi g a p odu t to a su ? Is the e a eas a to al ulate the a ea of a t ia gle if I a gi e its e ti es as poi ts o a Ca tesia pla e? Do ’t so e of the Pola g aphs i Chapte 9 look like the ha e ee d a ith a “pi og aph? Wh is that? A loid is oth a ra histo hrone a d a tauto hrone. What a e these a d h a e the i po ta t? ou ill ha e to look this o e up, ut it is ell o th ou ti e What is a e to oss p odu t a d ho is it used? Ho do the p ope ties of e to s e te d to di e sio s, he e the eall atte ? Additio all , ask ou self:     What t ig ide tities a I eate that I ha e ot et see ? What Pola g aphs a I eate essi g ith t ig fu tio s? What akes a p ett g aph i stead of o e that just looks essed up? Ca I o e up ith a si ple ethod of doi g thi gs tha I a ei g taught? What p o le s a I o e up ith to stu p f ie ds? Those ho app oa h ath i this Please feel f ee to o ta t a e ill e to o o ’s leade s. A e ou o e of the ? e at ea l@ athgu .us if ou ha e a Tha k ou a d est ishes! Ea l Version 2.1 uestio s o o e ts. Co e a t Re e a Willia s, T itte ha dle: @jolteo kitt Page 2 of 109 April 10, 2017 Trigonometry Handbook Table of Contents Page 7 9 9 9 9 9 1 11 11 11 1 1 14 1 17 19 4 4 Version 2.1 Descriptio Chapter : Fu ctio s a d Special A gles I troductio A gle Defi itio s Fu ctio Defi itio s o the x‐ a d y‐ A es Pythagorea Ide tities Si e‐Cosi e Relatio ship Key A gles i Radia s a d Degrees Cofu ctio s U it Circle Fu ctio Defi itio s i a Right Tria gle SOH‐CAH‐TOA Trigo o etric Fu ctio s of Special A gles Trigo o etric Fu ctio Values i Quadra ts II, III, a d IV Pro le s I volvi g Trig Fu ctio Values i Quadra ts II, III, a d IV Pro le s I volvi g A gles of Depressio a d I cli atio Chapter 2: Graphs of Trig Fu ctio s Basic Trig Fu ctio s Characteristics of Trigo o etric Fu ctio Graphs Ta le of Trigo o etric Fu ctio Characteristics Si e Fu ctio Cosi e Fu ctio Ta ge t Fu ctio Cota ge t Fu ctio Seca t Fu ctio Coseca t Fu ctio Applicatio : Si ple Har o ic Motio Chapter : I verse Trigo o etric Fu ctio s Defi itio s Pri cipal Values a d Ra ges Graphs of I verse Trig Fu ctio s Pro le s I volvi g I verse Trigo o etric Fu ctio s Page 3 of 109 April 10, 2017 Trigonometry Handbook Table of Contents Page Descriptio 41 41 41 4 Chapter : Key A gle For ulas A gle Additio , Dou le A gle, Half A gle For ulas E a ples Po er Reduci g For ulas Product‐to‐Su For ulas Su ‐to‐Product For ulas E a ples 4 44 47 4 Chapter : Trigo o etric Ide tities a d E uatio s Verifyi g Ide tities Verifyi g Ide tities ‐ Tech i ues Solvi g Trigo etic E uatio s Solvi g Trigo etic E uatio s ‐ E a ples 7 1 4 7 9 1 1 4 4 7 Version 2.1 Chapter : Solvi g a O li ue Tria gle Su ary of Methods La s of Si es a d Cosi es La s of Si es a d Cosi es ‐ E a ples The A iguous Case Flo chart for the A iguous Case A iguous Case ‐ E a ples Beari gs Beari gs ‐ E a ples Chapter 7: Area of a Tria gle Geo etry For ula Hero 's For ula Trigo o etric For ulas Coordi ate Geo etry For ula E a ples Chapter : Polar Coordi ates I troductio Co versio et ee Recta gular a d Polar Coordi ates E pressi g Co ple Nu ers i Polar For Operatio s o Co ple Nu ers i Polar For DeMoivre's Theore DeMoivre's Theore for Roots Page 4 of 109 April 10, 2017 Trigonometry Handbook Table of Contents Page 9 9 7 7 71 74 7 7 77 Descriptio Chapter 9: Polar Fu ctio s Parts of the Polar Graph Sy etry Graphi g Methods Graphi g ith the TI‐ 4 Plus Calculator Graph Types Circles, Roses, Li aço s, Le iscates, Spirals Rose Cardioid Co verti g Bet ee Polar a d Recta gular For s of E uatio s Para etric E uatio s 7 9 9 Chapter : Vectors I troductio Special U it Vectors Vector Co po e ts Vector Properties Vector Properties ‐ E a ples Dot Product Dot Product ‐ E a ples Vector Projectio Orthogo al Co po e ts of a Vector Work Applicatio s of Vectors – E a ples Vector Cross Product Vector Triple Products 9 1 1 Appe dices Appe di A ‐ Su ary of Trigo o etric For ulas Appe di B ‐ Solvi g The A iguous Case ‐ Alter ative Method Appe di C ‐ Su ary of Polar a d Recta gular For s 1 I dex 79 79 79 1 4 Version 2.1 Page 5 of 109 April 10, 2017 Trigonometry Handbook Table of Contents Useful We sites Mathguy.us – Developed specifically for ath stude ts fro Middle School to College, ased o the author's e te sive e perie ce i professio al athe atics i a usi ess setti g a d i ath tutori g. Co tai s free do loada le ha d ooks, PC Apps, sa ple tests, a d ore. http:// . athguy.us/ Wolfra Math World – Perhaps the pre ier site for athe atics o the We . This site co tai s defi itio s, e pla atio s a d e a ples for ele e tary a d adva ced ath topics. http:// ath orld. olfra .co / Kha Acade y – Supplies a free o li e collectio of thousa ds of icro lectures via YouTu e o u erous topics. It's ath a d scie ce li raries are e te sive. .kha acade y.org A alyze Math Trigo o etry – Co tai s free Trigo o etry tutorials a d pro le s. Uses Java applets to e plore i porta t topics i teractively. http:// .a alyze ath.co /Trigo o etry.ht l Schau ’s Outli e A i porta t stude t resource for a y high school or college ath stude t is a Schau ’s Outli e. Each ook i this series provides e pla atio s of the various topics i the course a d a su sta tial u er of pro le s for the stude t to try. Ma y of the pro le s are orked out i the ook, so the stude t ca see e a ples of ho they should e solved. Schau ’s Outli es are availa le at A azo .co , Bar es & No le a d other ooksellers. Version 2.1 Page 6 of 109 April 10, 2017 Chapter 1 Fu ctio s a d Special A gles I troductio What is Trigo o etry? The ord Trigo o etry co es fro the Greek trigo o ea i g tria gle a d etro ea i g easure . So, si ply put, Trigo o etry is the study of the easures of tria gles. This i cludes the le gths of the sides, the easures of the a gles a d the relatio ships et ee the sides a d a gles. The oder approach to Trigo o etry also deals ith ho right tria gles i teract ith circles, especially the U it Circle, i.e., a circle of radius 1. Although the asic co cepts are si ple, the applicatio s of Trigo o etry are far reachi g, fro cutti g the re uired a gles i kitche tiles to deter i i g the opti al trajectory for a rocket to reach the outer pla ets. Radia s a d Degrees A gles i Trigo o etry ca   e easured i either radia s or degrees: degrees i.e., ° i o e rotatio arou d a circle. Although there are various There are accou ts of ho a circle ca e to have degrees, ost of these are ased o the fact that early civilizatio s co sidered a co plete year to have days. There are ~ . radia s i o e rotatio arou d a circle. The a cie t Greeks defi ed to e the ratio of the circu fere ce of a circle to its dia eter i.e., . Si ce the dia eter is dou le the radius, the circu fere ce is ti es the radius i.e., . O e radia is the easure of the a gle ade fro rappi g the radius of a circle alo g the circle’s e terior. r 1 rad r Measure of a Arc O e of the si plest a d ost asic for ulas i Trigo o etry provides the easure of a arc i ter s of the radius of the circle, , a d the arc’s ce tral a gle θ, e pressed i radia s. The for ula is easily derived fro the portio of the circu fere ce su te ded y θ. Si ce there are radia s i o e full rotatio arou d the circle, the of a arc ith ce tral a gle θ, e pressed i radia s, is: ∙ Version 2.1 θ ∙ θ easure so Page 7 of 109 April 10, 2017 Chapter 1 Fu ctio s a d Special A gles A gle Defi itio s Basic Defi itio s A fe defi itio s relati g to a gles are useful he egi A gle: A easure of the space et ee rays ith a co easured y the a ou t of rotatio re uired to get fro side to its ter i al side. I itial Side: The side of a a gle fro easure egi s. i g the study of Trigo o etry. o e dpoi t. A a gle is typically its i itial hich its rotatio al Ter i al Side: The side of a a gle at hich its rotatio al easure e ds. Vertex: The verte of a a gle is the co Defi itio s i the Cartesia o e dpoi t of the t o rays that defi e the a gle. Pla e Whe a gles are graphed o a coordi ate syste Recta gular or Polar , a u er of additio al ter s are useful. Sta dard Positio : A a gle is i sta dard positio if its verte is the origi i.e., the poi t , a d its i itial side is the positive ‐a is. Polar Axis: The Polar A is is the positive ‐a is. It is the i itial side of all a gles i sta dard positio . Polar A gle: For a a gle i sta dard positio , its polar a gle is the a gle easured fro the polar a is to its ter i al side. If easured i a cou ter‐clock ise directio , the polar a gle is positive; if easured i a clock ise directio , the polar a gle is egative. Refere ce A gle: For a a gle i sta dard positio , its refere ce a gle is the a gle et ee ° a d 9 ° easured fro the ‐a is positive or egative to its ter i al side. The refere ce a gle ca e °; it ca e 9 °; it is ever egative. Coter i al A gle: T o a gles are coter i al if they are i sta dard positio a d have the sa e ter i al side. For e a ple, a gles of easure ° a d ° are coter i al ecause ° is o e full rotatio arou d the circle i.e., ° , plus °, so they have the sa e ter i al side. Quadra tal A gle: A a gle i sta dard positio is a uadra tal a gle if its ter i al side lies o either the ‐a is or the ‐a is. Version 2.1 Page 8 of 109 April 10, 2017 Chapter 1 Fu ctio s a d Special A gles Trigo o etric Fu ctio s Trigo o etric Fu ctio s o the ‐ a d ‐axes sin θ sin θ cos θ cos θ tan θ tan θ cot θ cot θ sec θ sec θ csc θ Si e‐Cosi e Relatio ship Pythagorea Ide tities for a y a gle θ sin cos sec csc sin θ sin θ tan cot Cofu ctio s i Quadra t I cos ⇔ cos sec csc ⇔ csc Version 2.1 cot ⇔ cos θ cos θ sec θ cot θ tan θ tan θ cot θ cot sin sin θ cos θ cos θ sin θ cos θ sin θ Key A gles ° ° radians ° radians ° ° sin tan csc θ csc θ 9 ° radians radians radians tan sec Page 9 of 109 April 10, 2017 Chapter 1 Fu ctio s a d Special A gles The U it Circle The U it Circle diagra elo provides ‐ a d ‐values o a circle of radius at key a gles. At a y poi t o the u it circle, the ‐coordi ate is e ual to the cosi e of the a gle a d the ‐coordi ate is e ual to the si e of the a gle. Usi g this diagra , it is easy to ide tify the si es a d cosi es of a gles that recur fre ue tly i the study of Trigo o etry. Version 2.1 Page 10 of 109 April 10, 2017 Chapter 1 Fu ctio s a d Special A gles Trigo o etric Fu ctio s a d Special A gles Trigo o etric Fu ctio s Right Tria gle SOH‐CAH‐TOA sin sin tan tan cos sin cos cos tan Special A gles Trig Fu ctio s of Special A gles Radia s Degrees ⁰ √ √ √ ⁰ √ √ √ ⁰ √ √ √ ⁰ √ √ 9 ⁰ √ √ Note the patter s i the a ove ta le: I the si e colu is the si e colu u der the radical! The cosi e colu Version 2.1 Page 11 of 109 √ √ √ √ √ √ √ undefined , the u ers to occur i se ue ce reversed. Ta ge t si e cosi e. April 10, 2017 Chapter 1 Fu ctio s a d Special A gles Trigo o etric Fu ctio Values i Quadra ts II, III, a d IV I uadra ts other tha Quadra t I, trigo o etric values for a gles are calculated i the follo i g a er:     Dra the a gle θ o the Cartesia Pla e. Calculate the easure of the refere ce a gle fro the ‐a is to θ. Fi d the value of the trigo o etric fu ctio of the a gle i the previous step. Assig a or sig to the trigo o etric value ased o the fu ctio used a d the uadra t θ is i fro the ta le at right . Exa ples: Θ i Quadra t II – Calculate: For sin Θ i Quadra t III – Calculate: For cos ° √ ∠ ⁰, the refere ce a gle is , so: ° √ ° ° ⁰ √ ⁰, the refere ce a gle is , so: ° ° tan Version 2.1 ° √ ∠ ° ° ° ° ° ° Θ i Quadra t IV – Calculate: For ⁰ ⁰ ⁰, the refere ce a gle is , so: Page 12 of 109 ° ° ∠ April 10, 2017 Chapter 1 Fu ctio s a d Special A gles Pro le s I volvi g Trig Fu ctio Values i Quadra ts II, III, a d IV A typical pro le i Trigo o etry is to fi d the value of o e or ore Trig fu ctio s ased o a set of co strai ts. Ofte , the co strai ts i volve the value of a other Trig fu ctio a d the sig of yet a third Trig Fu ctio . The key to solvi g this type of pro le is to dra the correct tria gle i the correct uadra t. A couple of e a ples ill illustrate this process. Exa ple . : sin Notice that sin I , . Fi d the values of sec , tan is egative; . Therefore, , tan is i a d cot . , so e dra the a gle i that uadra t. is al ays positive. Si ce sin e let , , Usi g the Pythagorea Theore , e calculate the le gth of the horizo tal leg of the tria gle: egative, so e √ . Si ce the a gle is i √ . ust have The , sec √ √ A d, cot Exa ple .2: cot Notice that cot I , is egative, a d √ . Therefore, , cos is √ . Fi d the value of csc , cos , is i is positive. Si ce cot a d cos . , so e dra the a gle i that uadra t. , e let Usi g the Pythagorea Theore , e ca calculate the le gth of the hypote use of the tria gle: √ The , csc A d, cos Version 2.1 √ . 9 9, . √97. √ Page 13 of 109 April 10, 2017 Chapter 1 Fu ctio s a d Special A gles Pro le s I volvi g A gles of Depressio a d I cli atio A co o pro le i Trigo o etry deals ith a gles of depressio or i cli atio . A a gle of depressio is a a gle elo the horizo tal at hich a o server ust look to see a o ject. A a gle of i cli atio is a a gle a ove the horizo tal at hich a o server ust look to see a o ject. Exa ple . : A uildi g feet tall casts a foot lo g shado . If a perso looks do fro the top of the uildi g, hat is the easure of the a gle of depressio ? Assu e the perso 's eyes are feet a ove the top of the uildi g. The total height fro hich the perso looks do upo the shado is: egi y dra i g the diagra elo , the co sider the trigo o etry i volved. tan x° tan 9 . . 9 ft. We 7. ° The a gle of depressio is the co ple e t of °. θ 9 ° 7. ° 7 . ° Exa ple . : A ship is eters fro a vertical cliff. The avigator uses a se ta t to deter i e the a gle of i cli atio fro the deck of the ship to the top of the cliff to e . °. Ho far a ove the deck of the ship is the top of the cliff? What is the dista ce fro the deck to the top of the cliff? We egi y dra i g the diagra a elo , the co sider the trigo o etry i volved. To fi d ho tan . ° far a ove the deck the top of the cliff is tan . ° To fi d the dista ce fro cos . ° . ° Version 2.1 7. Page 14 of 109 7 . : eters the deck to the top of the cliff : eters April 10, 2017 Chapter Graphs of Trig Fu ctio s Graphs of Basic Pare t Trigo o etric Fu ctio s The si e a d coseca t fu ctio s are reciprocals. So: sin csc and csc sin The cosi e a d seca t fu ctio s are reciprocals. So: cos sec and sec cos The ta ge t a d cota ge t fu ctio s are reciprocals. So: tan Version 2.1 cot and cot Page 15 of 109 tan April 10, 2017 Chapter Graphs of Trig Fu ctio s Graphs of Basic Pare t Trigo o etric Fu ctio s It is i structive to vie the pare t trigo o etric fu ctio s o the sa e a es as their reciprocals. Ide tifyi g patter s et ee the t o fu ctio s ca e helpful i graphi g the . Looki g at the si e a d coseca t fu ctio s, e see that they i tersect at their a i u a d i i u values i.e., he . The vertical asy ptotes ot sho of the coseca t fu ctio occur he the si e fu ctio is zero. Looki g at the cosi e a d seca t fu ctio s, e see that they i tersect at their a i u a d i i u values i.e., he . The vertical asy ptotes ot sho of the seca t fu ctio occur he the cosi e fu ctio is zero. Looki g at the ta ge t a d cota ge t fu ctio s, e see that they i tersect he sin cos i.e., at , a i teger . The vertical asy ptotes ot sho of the each fu ctio occur he the other fu ctio is zero. Version 2.1 Page 16 of 109 April 10, 2017 Chapter Graphs of Trig Fu ctio s Characteristics of Trigo o etric Fu ctio Graphs All trigo o etric fu ctio s are periodic, ea i g that they repeat the patter of the curve called a cycle o a regular asis. The key characteristics of each curve, alo g ith k o ledge of the pare t curves are sufficie t to graph a y trigo o etric fu ctio s. Let’s co sider the ge eral fu ctio : here A, B, C a d D are co sta ts a d ta ge t, cota ge t, seca t, coseca t . A∙ B C D is a y of the si trigo o etric fu ctio s si e, cosi e, A plitude A plitude is the easure of the dista ce of peaks a d troughs fro the idli e i.e., ce ter of a sine or cosine function; a plitude is al ays positive. The other four fu ctio s do ot have peaks a d troughs, so they do ot have a plitudes. For |A|. the ge eral fu ctio , , defi ed a ove, amplitude Period Period is the horizo tal idth of a si gle cycle or ave, i.e., the dista ce it travels efore it repeats. Every trigo o etric fu ctio has a period. The periods of the parent functions are as follo s: for si e, cosi e, seca t a d coseca t, period π; for ta ge t a d cota ge t, period π. For the ge eral fu ctio , , defi ed a ove, . period Fre ue cy Fre ue cy is ost useful he used ith the si e a d cosi e fu ctio s. It is the reciprocal of the period, i.e., frequency . Fre ue cy is typically discussed i relatio to the si e a d cosi e fu ctio s he co sideri g har o ic otio or aves. I Physics, fre ue cy is typically easured i Hertz, i.e., cycles per seco d. 1 Hz 1 cycle per seco d. For the ge eral si e or cosi e fu ctio , Version 2.1 , defi ed a ove, frequency Page 17 of 109 . April 10, 2017 Chapter Graphs of Trig Fu ctio s Phase Shift Phase shift is ho far has the fu ctio ee shifted horizo tally left or right fro its pare t fu ctio . For the ge eral fu ctio , , defi ed a ove, phase shift . A positive phase shift i dicates a shift to the right relative to the graph of the pare t fu ctio ; a egative phase shift i dicates a shift to the left relative to the graph of the pare t fu ctio . A trick for calculati g the phase shift is to set the argu e t of the trigo o etric fu ctio e ual to zero: B C , a d solve for . The resulti g value of is the phase shift of the fu ctio . Vertical Shift Vertical shift is the vertical dista ce that the idli e of a curve lies a ove or elo the idli e of its pare t fu ctio i.e., the ‐a is . For the ge eral fu ctio , , defi ed a ove, vertical shift D. The value of D ay e positive, i dicati g a shift up ard, or egative, i dicati g a shift do ard relative to the graph of the pare t fu ctio . Putti g it All Together The illustratio Version 2.1 elo sho s ho all of the ite s descri ed a ove co Page 18 of 109 i e i a si gle graph. April 10, 2017 Chapter Su Fu ctio : Graphs of Trig Fu ctio s ary of Characteristics a d Key Poi ts – Trigo o etric Fu ctio Graphs Si e Cosi e Ta ge t Cota ge t Seca t Coseca t Pare t Fu ctio sin cos tan cot sec csc Do ai ∞, ∞ ∞, ∞ Vertical Asy ptotes o e o e , , Ra ge Period ‐i tercepts Odd or Eve Fu ctio 1 , here is a I teger Ge eral For A plitude/Stretch, Period, Phase Shift, Vertical Shift 2 whe Odd Fu ctio sin | |, , here here , is odd Eve Fu ctio cos , , | |, , ∞, ∞ e cept ∞, ∞ e cept , here is a I teger is odd here , here I teger is odd ∞, ∞ id ay et ee asy ptotes id ay et ee asy ptotes Odd Fu ctio Odd Fu ctio tan , , | |, cot , , , ∞, here ∪ ∞, ∞ e cept , here is a I teger , here is a is odd ,∞ I teger ∞, o e Eve Fu ctio | |, ∪ ,∞ o e Odd Fu ctio csc , | |, , vertical asy ptote whe , , vertical asy ptote vertical asy ptote whe vertical asy ptote vertical asy ptote whe vertical asy ptote whe Notes: 1 , is odd sec | |, , here is a ∞, ∞ , ∞, ∞ e cept vertical asy ptote A odd fu ctio is sy etric a out the origi , i.e. . A eve fu ctio is sy etric a out the ‐a is, i.e., All Phase Shifts are defi ed to occur relative to a starti g poi t of the ‐a is i.e., the vertical li e Version 2.1 Page 19 of 109 vertical asy ptote . . April 10, 2017 Chapter Graphs of Trig Fu ctio s Graph of a Ge eral Si e Fu ctio Ge eral For The ge eral for . of a si e fu ctio is: I this e uatio , e fi d several para eters of the fu ctio   A plitude: fu ctio fro Period: | |. The a plitude is the its pare t fu ctio : sin . hich ill help us graph it. I particular: ag itude of the stretch or co pressio of the . The period of a trigo o etric fu ctio is the horizo tal dista ce over hich the curve travels efore it egi s to repeat itself i.e., egi s a e cycle . For a si e or cosi e fu ctio , this is the le gth of o e co plete ave; it ca e easured fro peak to peak or sin . fro trough to trough. Note that π is the period of  Phase Shift: . The phase shift is the dista ce of the horizo tal tra slatio of the fu ctio . Note that the value of i the ge eral for has a i us sig i fro t of it, just like does i the verte for of a uadratic e uatio : . So, o A i us sig i fro t of the i plies a tra slatio to the right, a d o A plus sig i fro t of the i plies a i plies a tra slatio to the left.  Vertical Shift: e uivale t to . This is the dista ce of the vertical tra slatio of the fu ctio . This is i the verte for of a uadratic e uatio : . Exa ple 2. : The idli e has the e uatio y D. I this e a ple, the idli e is: y . O e ave, shifted to the right, is sho i ora ge elo . For this exa ple: ; A plitude: Period: ; | | ; | | Phase Shift: Vertical Shift: Version 2.1 Page 20 of 109 April 10, 2017 Chapter Graphs of Trig Fu ctio s Graphi g a Si e Fu ctio with No Vertical Shift: A ave cycle of the si e fu ctio has three zero poi ts poi ts o the ‐a is – at the egi i g of the period, at the e d of the period, a d half ay i ‐ et ee . Step : Phase Shift: . . . The first ave egi s at the poi t u its to the right of the Origi . Step 2: Period: Exa ple: The poi t is: . The first . ave e ds at the poi t: The first ave e ds at the poi t u its to the right of where the wave egi s. , Step : The third zero poi t is located half ay et ee the first t o. The poi t is: Step : The ‐value of the poi t half ay et ee the left a d ce ter zero poi ts is " ". The poi t is: Step : The ‐value of the poi t half ay et ee the ce ter a d right zero poi ts The poi t is: is – , , , , . Step : Dra a s ooth curve through the five key poi ts. , , , This ill produce the graph of o e ave of the fu ctio . Step 7: Duplicate the ave to the left a d right as desired. Version 2.1 , Note: If , all poi ts o the curve are shifted vertically y u its. Page 21 of 109 April 10, 2017 Chapter Graphs of Trig Fu ctio s Graph of a Ge eral Cosi e Fu ctio Ge eral For The ge eral for . of a cosi e fu ctio is: I this e uatio , e fi d several para eters of the fu ctio   A plitude: fu ctio fro Period: | |. The a plitude is the its pare t fu ctio : cos . hich ill help us graph it. I particular: ag itude of the stretch or co pressio of the . The period of a trigo o etric fu ctio is the horizo tal dista ce over hich the curve travels efore it egi s to repeat itself i.e., egi s a e cycle . For a si e or cosi e fu ctio , this is the le gth of o e co plete ave; it ca e easured fro peak to peak or cos . fro trough to trough. Note that π is the period of  Phase Shift: . The phase shift is the dista ce of the horizo tal tra slatio of the fu ctio . Note that the value of i the ge eral for has a i us sig i fro t of it, just like does i the verte for of a uadratic e uatio : . So, o A i us sig i fro t of the i plies a tra slatio to the right, a d o A plus sig i fro t of the i plies a i plies a tra slatio to the left.  Vertical Shift: e uivale t to . This is the dista ce of the vertical tra slatio of the fu ctio . This is i the verte for of a uadratic e uatio : . Exa ple 2.2: The idli e has the e uatio y D. I this e a ple, the idli e is: y . O e ave, shifted to the right, is sho i ora ge elo . For this exa ple: ; A plitude: Period: ; | | ; | | Phase Shift: Vertical Shift: Version 2.1 Page 22 of 109 April 10, 2017 Chapter Graphs of Trig Fu ctio s Graphi g a Cosi e Fu ctio with No Vertical Shift: A ave cycle of the cosi e fu ctio has t o a i a or i i a if – o e at the egi i g of the period a d o e at the e d of the period – a d a i i u or a i u if half ay i ‐ et ee . Step : Phase Shift: . . , The first ave egi s at the poi t u its to the right of the poi t , . Step 2: Period: Exa ple: The poi t is: . The first . ave e ds at the poi t: The first ave e ds at the poi t u its to the right of where the wave egi s. , Step : The ‐value of the poi t half ay et ee those i the t o steps a ove is " ". The poi t is: Step : The ‐value of the poi t half ay et ee the left a d ce ter e tre a is " ". The poi t is: Step : The ‐value of the poi t half ay et ee the ce ter a d right e tre a is " ". The poi t is: , , , , Step : Dra a s ooth curve through the five key poi ts. , , , This ill produce the graph of o e ave of the fu ctio . Step 7: Duplicate the ave to the left a d right as desired. Version 2.1 , Note: If , all poi ts o the curve are shifted vertically y u its. Page 23 of 109 April 10, 2017 Chapter Graphs of Trig Fu ctio s Graph of a Ge eral Ta ge t Fu ctio Ge eral For The ge eral for . of a ta ge t fu ctio is: I this e uatio , e fi d several para eters of the fu ctio   hich ill help us graph it. I particular: Scale factor: | |. The ta ge t fu ctio does ot have a plitude. | | is the stretch or co pressio of the fu ctio fro its pare t fu ctio : tan . Period: ag itude of the . The period of a trigo o etric fu ctio is the horizo tal dista ce over hich the curve travels efore it egi s to repeat itself i.e., egi s a e cycle . For a ta ge t or cota ge t fu ctio , this is the horizo tal dista ce et ee co secutive asy ptotes it is also the dista ce et ee ‐i tercepts . Note that π is the period of tan .  Phase Shift: . The phase shift is the dista ce of the horizo tal tra slatio of the fu ctio . Note that the value of i the ge eral for has a i us sig i fro t of it, just like does i the verte for of a uadratic e uatio : . So, o A i us sig i fro t of the i plies a tra slatio to the right, a d o A plus sig i fro t of the i plies a i plies a tra slatio to the left.  Vertical Shift: e uivale t to . This is the dista ce of the vertical tra slatio of the fu ctio . This is i the verte for of a uadratic e uatio : . Exa ple 2. : The idli e has the e uatio y D. I this e a ple, the idli e is: y . O e cycle, shifted to the right, is sho i ora ge elo . Note that, for the ta ge t curve, e typically graph half of the pri cipal cycle at the poi t of the phase shift, a d the fill i the other half of the cycle to the left see e t page . Version 2.1 For this exa ple: ; ; Scale Factor: | | Period: | | ; Phase Shift: Vertical Shift: Page 24 of 109 April 10, 2017 Chapter Graphs of Trig Fu ctio s Graphi g a Ta ge t Fu ctio with No Vertical Shift: A cycle of the ta ge t fu ctio has t o asy ptotes a d a zero poi t half ay i ‐ et ee . It flo s up ard to the right if a d do ard to the right if Step : Phase Shift: . . The poi t is: . , . . Place a vertical asy ptote The right asy ptote is at: u its to the right of the egi . . The first cycle egi s at the zero poi t u its to the right of the Origi . Step 2: Period: Exa ple: i g of the cycle. Step : Place a vertical asy ptote left of the egi cycle. The left asy ptote is at: u its to the i g of the Step : The ‐value of the poi t half ay et ee the zero poi t a d the right asy ptote is " ". The poi t is: Step : The ‐value of the poi t half ay et ee the left asy ptote a d the zero poi t is " ". The poi t is: , , Step : Dra a s ooth curve through the three key poi ts, approachi g the asy ptotes o each side. , This ill produce the graph of o e cycle of the fu ctio . Step 7: Duplicate the cycle to the left a d right as desired. Version 2.1 , Note: If , all poi ts o the curve are shifted vertically y u its. Page 25 of 109 April 10, 2017 Chapter Graphs of Trig Fu ctio s Graph of a Ge eral Cota ge t Fu ctio Ge eral For The ge eral for . of a cota ge t fu ctio is: I this e uatio , e fi d several para eters of the fu ctio   hich ill help us graph it. I particular: Scale factor: | |. The cota ge t fu ctio does ot have a plitude. | | is the ag itude of the stretch or co pressio of the fu ctio fro its pare t fu ctio : cot . Period: . The period of a trigo o etric fu ctio is the horizo tal dista ce over hich the curve travels efore it egi s to repeat itself i.e., egi s a e cycle . For a ta ge t or cota ge t fu ctio , this is the horizo tal dista ce et ee co secutive asy ptotes it is also the dista ce et ee ‐i tercepts . Note that π is the period of cot .  Phase Shift: . The phase shift is the dista ce of the horizo tal tra slatio of the fu ctio . Note that the value of i the ge eral for has a i us sig i fro t of it, just like does i the verte for of a uadratic e uatio : . So, o A i us sig i fro t of the i plies a tra slatio to the right, a d o A plus sig i fro t of the i plies a i plies a tra slatio to the left.  Vertical Shift: e uivale t to . This is the dista ce of the vertical tra slatio of the fu ctio . This is i the verte for of a uadratic e uatio : . Exa ple 2. : The idli e has the e uatio y D. I this e a ple, the idli e is: y . O e cycle, shifted to the right, is sho i ora ge elo . Note that, for the cota ge t curve, e typically graph the asy ptotes first, a d the graph the curve et ee the see e t page . For this exa ple: ; ; Scale Factor: | | Period: | | ; Phase Shift: Vertical Shift: Version 2.1 Page 26 of 109 April 10, 2017 Chapter Graphs of Trig Fu ctio s Graphi g a Cota ge t Fu ctio with No Vertical Shift: A cycle of the cota ge t fu ctio has t o asy ptotes a d a zero poi t half ay i ‐ et ee . It flo s do ard to the right if a d up ard to the right if . Step : Phase Shift: . . . The left Place a vertical asy ptote u its to the right of the ‐axis. Step 2: Period: Exa ple: asy ptote is at: . . Place a other vertical asy ptote u its to the right of the first o e. The right asy ptote is at: Step : A zero poi t e ists half ay et ee the t o asy ptotes. , Step : The ‐value of the poi t half ay et ee the left asy ptote a d the zero poi t is " ". The poi t is: Step : The ‐value of the poi t half ay et ee the zero poi t a d the right asy ptote is " ". The poi t is: , , Step : Dra a s ooth curve through the three key poi ts, approachi g the asy ptotes o each side. , , This ill produce the graph of o e cycle of the fu ctio . Step 7: Duplicate the cycle to the left a d right as desired. Version 2.1 , The poi t is: Note: If , all poi ts o the curve are shifted vertically y u its. Page 27 of 109 April 10, 2017 Chapter Graphs of Trig Fu ctio s Graph of a Ge eral Seca t Fu ctio Ge eral For The ge eral for . of a seca t fu ctio is: I this e uatio , e fi d several para eters of the fu ctio   hich ill help us graph it. I particular: Scale factor: | |. The seca t fu ctio does ot have a plitude. | | is the ag itude of the stretch or co pressio of the fu ctio fro its pare t fu ctio : sec . Period: . The period of a trigo o etric fu ctio is the horizo tal dista ce over hich the curve travels efore it egi s to repeat itself i.e., egi s a e cycle . For a seca t or coseca t fu ctio , this is the horizo tal dista ce et ee co secutive a i a or i i a it is sec . also the dista ce et ee every seco d asy ptote . Note that π is the period of  Phase Shift: . The phase shift is the dista ce of the horizo tal tra slatio of the fu ctio . Note that the value of i the ge eral for has a i us sig i fro t of it, just like does i the verte for of a uadratic e uatio : . So, o A i us sig i fro t of the i plies a tra slatio to the right, a d o A plus sig i fro t of the i plies a i plies a tra slatio to the left.  Vertical Shift: e uivale t to . This is the dista ce of the vertical tra slatio of the fu ctio . This is i the verte for of a uadratic e uatio : . Exa ple 2. : The idli e has the e uatio y D. I this e a ple, the idli e is: y . O e cycle, shifted to the right, is sho i ora ge elo . O e cycle of the seca t curve co tai s t o U‐shaped curves, o e ope i g up a d o e ope i g do . For this exa ple: ; ; Scale Factor: | | Period: | | ; Phase Shift: Vertical Shift: Version 2.1 Page 28 of 109 April 10, 2017 Chapter Graphs of Trig Fu ctio s Graphi g a Seca t Fu ctio with No Vertical Shift: A cycle of the seca t fu ctio ca correspo di g cosi e fu ctio     e developed y first plotti g a cycle of the ecause sec The cosi e fu ctio ’s zero poi ts produce asy Ma i a for the cosi e fu ctio produce i i Mi i a for the cosi e fu ctio produce a i Seca t curves are U‐shaped, alter ately ope i . ptotes for the seca t fu ctio . a for the seca t fu ctio . a for the seca t fu ctio . g up a d ope i g do . Exa ple: . Step : Graph o e ave of the correspo di g cosi e fu ctio . The e uatio of the correspo di g cosi e fu ctio for the e a ple is: Step 2: Asy ptotes for the seca t fu ctio occur at the zero poi ts of the cosi e fu ctio . The zero poi ts occur at: Step : Each the cosi e fu represe ts a the seca t fu a i u of ctio i i u for ctio . Cosi e a i a a d, therefore, seca t i i a are Step : Each the cosi e fu represe ts a the seca t fu i i u ctio a i u ctio . The cosi e i i u a d, therefore, the seca t , Seca t asy ptotes are: at: of for a d , a i u Step : Dra s ooth U‐ shaped curves through each key poi t, approachi g the asy ptotes o each side. a d is at: , , This ill produce the graph of o e cycle of the fu ctio . Step : Duplicate the cycle to the left a d right as desired. Erase the cosi e fu ctio if ecessary. Version 2.1 a d , Note: If , all poi ts o the curve are shifted vertically y u its. Page 29 of 109 April 10, 2017 Chapter Graphs of Trig Fu ctio s Graph of a Ge eral Coseca t Fu ctio Ge eral For The ge eral for . of a coseca t fu ctio is: I this e uatio , e fi d several para eters of the fu ctio   hich ill help us graph it. I particular: Scale factor: | |. The coseca t fu ctio does ot have a plitude. | | is the ag itude of the stretch or co pressio of the fu ctio fro its pare t fu ctio : csc . Period: . The period of a trigo o etric fu ctio is the horizo tal dista ce over hich the curve travels efore it egi s to repeat itself i.e., egi s a e cycle . For a seca t or coseca t fu ctio , this is the horizo tal dista ce et ee co secutive a i a or i i a it is csc . also the dista ce et ee every seco d asy ptote . Note that π is the period of  Phase Shift: . The phase shift is the dista ce of the horizo tal tra slatio of the fu ctio . Note that the value of i the ge eral for has a i us sig i fro t of it, just like does i the verte for of a uadratic e uatio : . So, o A i us sig i fro t of the i plies a tra slatio to the right, a d o A plus sig i fro t of the i plies a i plies a tra slatio to the left.  Vertical Shift: e uivale t to . This is the dista ce of the vertical tra slatio of the fu ctio . This is i the verte for of a uadratic e uatio : . Exa ple 2. : The idli e has the e uatio y D. I this e a ple, the idli e is: y . O e cycle, shifted to the right, is sho i ora ge elo . O e cycle of the coseca t curve co tai s t o U‐shaped curves, o e ope i g up a d o e ope i g do . For this exa ple: ; ; Scale Factor: | | Period: | | ; Phase Shift: Vertical Shift: Version 2.1 Page 30 of 109 April 10, 2017 Chapter Graphs of Trig Fu ctio s Graphi g a Coseca t Fu ctio with No Vertical Shift: A cycle of the coseca t fu ctio ca correspo di g si e fu ctio     e developed y first plotti g a cycle of the ecause csc . The si e fu ctio ’s zero poi ts produce asy ptotes for the coseca t fu ctio . Ma i a for the si e fu ctio produce i i a for the coseca t fu ctio . Mi i a for the si e fu ctio produce a i a for the coseca t fu ctio . Coseca t curves are U‐shaped, alter ately ope i g up a d ope i g do . Exa ple: . Step : Graph o e ave of the correspo di g si e fu ctio . The e uatio of the correspo di g si e fu ctio for the e a ple is: Step 2: Asy ptotes for the coseca t fu ctio occur at the zero poi ts of the si e fu ctio . The zero poi ts occur at: Step : Each a i u of the si e fu ctio represe ts a i i u for the coseca t fu ctio . The si e a i u a d, therefore, the coseca t i i u is at: , Step : Each i i u of the si e fu ctio represe ts a a i u for the coseca t fu ctio . The si e i i u a d, therefore, the coseca t , , , , Coseca t asy ptotes are: a i u Step : Dra s ooth U‐ shaped curves through each key poi t, approachi g the asy ptotes o each side. , is at: , , This ill produce the graph of o e cycle of the fu ctio . Step : Duplicate the cycle to the left a d right as desired. Erase the si e fu ctio if ecessary. Version 2.1 , Note: If , all poi ts o the curve are shifted vertically y u its. Page 31 of 109 April 10, 2017 Chapter Graphs of Trig Fu ctio s Si ple Har o ic Motio otio I Physics, Si ple Har o ic Motio is a oscillati g otio thi k: repeati g up a d do here the force applied to a o ject is proportio al to a d i the opposite directio of its displace e t. A co o e a ple is the actio of a coiled spri g, hich oscillates up a d do he released. Such otio ca e odeled y the si e a d cosi e fu ctio s, usi g the follo i g e uatio s ote: is the lo er case Greek letter o ega, ot the E glish letter w : Har o ic otio e uatio s: Period: Fre ue cy: or cos or sin with Situatio s i hich a o ject starts at rest at the ce ter of its oscillatio , or at rest, use the si e fu ctio ecause sin ; situatio s i hich a o ject starts i a up or do positio prior to its release use the cosi e fu ctio ecause cos . Exa ple 2.7: A o ject is attached to a coiled spri g. The o ject is pulled up a d the released. If the a plitude is c a d the period is 7 seco ds, rite a e uatio for the dista ce of the o ject fro its starti g positio after seco ds. The spri g ill start at a ‐value of si ce it is pulled up , a d oscillate et ee a d a se t a y other force over ti e. A good represe tatio of this ould e a cosi e curve ith lead coefficie t . The period of the fu ctio is 7 seco ds. So, e get: period 7 and The resulti g e uatio , the , is: cos ∙ 7 7 Exa ple 2. : A o ject i si ple har o ic otio has a fre ue cy of . oscillatio s per seco d a d a a plitude of 1 c . Write a e uatio for the dista ce of the o ject fro its rest positio after seco ds. Assu i g that dista ce at ti e , it akes se se to use a si e fu ctio for this pro le . Si ce the a plitude is c , a good represe tatio of this ould e a si e curve ith lead coefficie t . Note that a lead coefficie t ould ork as ell. Recalli g that , ith The resulti g e uatio s, the , are: Version 2.1 . e get: sin Page 32 of 109 ∙ . or . sin April 10, 2017 Chapter I verse Trigo o etric Fu ctio s I verse Trigo o etric Fu ctio s I verse Trigo o etric Fu ctio s I verse trigo o etric fu ctio s are sho ith a " " e po e t or a arc prefi . So, the i verse si e of ay e sho as sin or arcsin . I verse trigo o etric fu ctio s ask the uestio : hich a gle has a fu ctio value of ? For e a ple: sin arctan . asks hich a gle has a si e value of . . It is e uivale t to: sin . . asks hich a gle has a ta ge t value of 1. It is e uivale t to: tan . Pri cipal Values of I verse Trigo o etric Fu ctio s There are a i fi ite u er of a gles that a s er the a ove uestio s, so the i verse trigo o etric fu ctio s are referred to as ulti‐valued fu ctio s. Because of this, athe aticia s have defi ed a pri cipal solutio for pro le s i volvi g i verse trigo o etric fu ctio s. The a gle hich is the pri cipal solutio or pri cipal value is defi ed to e the solutio that lies i the uadra ts ide tified i the figure at right. For e a ple: The solutio s to the e uatio i tervals sin . are all ‐values i the . That is, the set of all ∪ solutio s to this e uatio co tai s the t o solutio s i the i terval , , as ell as all a gles that are i teger ultiples of less tha or greater tha those t o a gles. Give the co fusio this ca create, defi ed a pri cipal value for the solutio to these ki ds of e uatio s. The pri cipal value of for hich . lies i Q1 ecause . is positive, a d is sin Ra ges of I verse Trigo o etric Fu ctio s . Ra ges of I verse Trigo o etric Fu ctio s The ra ges of i verse trigo o etric fu ctio s are ge erally defi ed to e the ra ges of the pri cipal values of those fu ctio s. A ta le su arizi g these is provided at right. Fu ctio A gles i Q4 are e pressed as egative a gles. Version 2.1 athe aticia s have Page 33 of 109 Ra ge sin cos tan April 10, 2017 Chapter I verse Trigo o etric Fu ctio s Graphs of I verse Trigo o etric Fu ctio s Pri cipal values are sho Version 2.1 Page 34 of 109 i gree . April 10, 2017 Chapter I verse Trigo o etric Fu ctio s Pro le s I volvi g I verse Trigo o etric Fu ctio s or It is te pti g to elieve, for e a ple, that sin sin . The t o fu ctio s are, after all i verses. Ho ever, tan tan this is ot al ays the case ecause the i verse fu ctio value desired is typically its pri cipal value, hich the stude t ill recall is defi ed o ly i certai uadra ts see the ta le at right . Let’s look at a couple of pro le s to see ho they are solved. Exa ple . : Calculate the pri cipal value of tan Begi y otici g that tan solutio to this pro le tan . a d tan are i verse fu ctio s, so the is related to the a gle give : . This a gle is i Q , ut the i verse ta ge t fu ctio is defi ed o ly i Q1 a d Q4, o the i terval , . We seek the a gle i Q1 or Q4 that has the sa e ta ge t value as . Si ce the ta ge t fu ctio has period , e ca calculate: tan i Q4 as our solutio . tan Exa ple .2: Calculate the pri cipal value of sin cos . We are looki g for the a gle hose si e value is cos i the i terval Method 2: Recall: sin θ sin Method : sin cos The , sin cos sin sin sin sin sin √ si ce si e values are egative i Q4. cos θ. The , cos ecause cos ecause . , ≡ sin a d sin sin . is i the i terval ecause i verse fu ctio s ork icely i . , uadra ts i hich the pri cipal values of the i verse fu ctio s are defi ed. Version 2.1 Page 35 of 109 April 10, 2017 Chapter I verse Trigo o etric Fu ctio s Pro le s I volvi g I verse Trigo o etric Fu ctio s Whe the i verse trigo o etric fu ctio is the i er fu ctio i a co positio of fu ctio s, it ill usually e ecessary to dra a tria gle to solve the pro le . I these cases, dra the tria gle defi ed y the i er i verse trig fu ctio . The derive the value of the outer trig fu ctio . √ Exa ple . : Calculate the value of cot sin Recall that the argu e t of the sin . √ fu ctio , . Dra the tria gle ased o this. Ne t, calculate the value of the tria gle’s horizo tal leg: √ √ . Based o the diagra , the , √ cot sin Exa ple . : Calculate the value of tan cos Recall that the argu e t of the cos √ fu ctio , √ √ . √ . Dra the tria gle ased o this. Ne t, calculate the value of the tria gle’s vertical leg: √ . √ Based o the diagra , the , √ tan cos fu ctio , √ √ Exa ple . : Calculate a alge raic e pressio for sin sec Recall that the argu e t of the sec √ √ . . Dra the tria gle ased o this. Ne t, calculate the value of the tria gle’s vertical leg: √ 9 Based o the diagra , the , sin sec Version 2.1 Page 36 of 109 √ 9 √ 9 √ April 10, 2017 Chapter 4 Key A gle For ulas Key A gle For ulas A gle Additio For ulas sin sin sin cos sin cos tan cos sin cos sin cos cos cos cos cos cos sin sin sin sin tan Dou le A gle For ulas sin sin cos cos tan Half A gle For ulas cos cos sin The use of a + or ‐ sin sig i the half a gle for ulas depe ds o the uadra t i sin the a gle resides. See chart elo . Sig s of Trig Fu ctio s By Quadra t cos tan Version 2.1 hich Page 37 of 109 si + cos ‐ ta ‐ si + cos + ta + si ‐ cos ‐ ta + si ‐ cos + ta – April 10, 2017 Chapter 4 Key A gle For ulas Key A gle For ulas – Exa ples Exa ple . : Fi d the e act value of: cos 7 ˚ cos Recall: cos cos 7 ˚ cos cos cos ˚ sin sin sin 7 ˚ sin ˚ ˚ cos 7 ˚ cos cos Exa ple .2: Fi d the e act value of: tan tan ° tan ° ° ° ° √ √ ° °∙ √ ∙ √ ° sin sin √ ° ∙ √ ∙ √ Version 2.1 √ Recall: tan ∙ Co verti g to Q1 a gles ° √ √ √ √ ° . Recall: sin Note: oth a gles are i Q1, hich ° ° ∙ cos Co verti g to a a gle i Q1 ˚ A gles i Q4 a d Q1 Exa ple . : Fi d the e act value of: sin sin ˚ √ √ ∙ ° √ or ° ˚ . ˚ ° ° ° ∙ sin 7 ˚ sin ∙ sin √ ° ∙ cos sin cos sin cos akes thi gs easier. ° √ Page 38 of 109 April 10, 2017 Chapter 4 Key A gle For ulas Exa ple . : sin , lies i uadra t II, a d cos , lies i uadra t I. Fi d cos Co struct tria gles for the t o a gles, ei g careful to co sider the sig s of the values i each uadra t: The , cos cos ∙ Exa ple . : Give the diagra tan ∙ 7 7 Exa ple . : tan cos √ √ ∙ 7 , a d ∙ sin at right, fi d: tan 7 7 sin lies i uadra t III. Find sin , cos Dra the tria gle elo , the apply the appropriate for ulas. sin cos tan Version 2.1 sin cos cos sin cos sin Page 39 of 109 ∙ 7 7 ∙ , tan 7 . 7 April 10, 2017 . Chapter 4 Key A gle For ulas Exa ple .7: Fi d the e act value of: cos Note that cos cos is i Q1, so the value of cos Recall: cos is positive. Usi g the half‐a gle for ula a ove Co verti g to a a gle i Q1 √ Exa ple . : csc Note that if cos sin , √ uadra t IV. Fi d sin . is i Q , so the value of sin cos √ Recall: sin is positive. Note: cosi e is positive i Q4 sin Usi g the half‐a gle for ula a ove –√ √ Version 2.1 lies i is i Q4, the so, sin sin √ √ ∙ √ √ Page 40 of 109 April 10, 2017 Chapter 4 Key A gle For ulas Key A gle For ulas Power Reduci g For ulas sin cos tan Product‐to‐Su For ulas ∙ ∙ ∙ ∙ Su ‐to‐Product For ulas ∙ ∙ ∙ ∙ ∙ ∙ ∙ Version 2.1 ∙ Page 41 of 109 April 10, 2017 Chapter 4 Key A gle For ulas Key A gle For ulas – Exa ples Exa ple .9: Co vert to a su Use: sin ∙ ∙ cos Exa ple . Use: cos ∙ cos sin sin : Co vert to a su for ula: cos ∙ ∙ cos Exa ple . for ula: sin cos cos Use: : Co vert to a product for ula: sin sin sin ∙ ∙ sin ∙ Use: Version 2.1 cos ∙ ∙ cos ∙ ∙ sin ∙ sin ∙ Exa ple . 2: Co vert to a product for ula: cos cos ∙ cos ∙ ∙ cos ∙ sin Page 42 of 109 April 10, 2017 Chapter Ide tities a d E uatio s Verifyi g Ide tities A sig ifica t portio of a y trigo o etry course deals ith verifyi g Trigo o etric Ide tities, i.e., state e ts that are al ays true assu i g the trigo o etric values i volved e ist . This sectio deals ith ho the stude t ay approach verificatio of ide tities such as: tan ∙ sin I verifyi g a Trigo o etric Ide tity, the stude t is asked to ork ith o ly o e side of the ide tity a d, usi g the sta dard rules of athe atical a ipulatio , derive the other side. The stude t ay ork ith either side of the ide tity, so ge erally it is est to ork o the side that is ost co ple . The steps elo prese t a strategy that ay e useful i verifyi g ide tities. Verificatio Steps 1. Ide tify which side you wa t to work o . Let’s call this Side A. Let’s call the side you are ot orki g o Side B. So, you ill e orki g o Side A to ake it look like Side B. a. If o e side has a ultiple of a a gle e.g., tan a d the other side does ot e.g., cos , ork ith the side that has the ultiple of a a gle. . If o e side has o ly si es a d cosi es a d the other does ot, ork ith the side that does ot have o ly si es a d cosi es. c. If you get part ay through the e ercise a d realize you should have started ith the other side, start over a d ork ith the other side. . If ecessary, i vestigate Side B y orki g o it a little. This is ot a violatio of the rules as lo g as, i your verificatio , you co pletely a ipulate Side A to look like Side B. If you choose to i vestigate Side B, ove your ork off a little to the side so it is clear you are i vestigati g a d ot actually orki g side B. . Si plify Side A as uch as possi le, ut re e er to look at the other side to ake sure you are ovi g i that directio . Do this also at each step alo g the ay, as lo g as it akes Side A look ore like Side B. a. Use the Pythagorea Ide tities to si plify, e.g., if o e side co tai s sin a d the other side co tai s cosi es ut ot si es, replace sin ith cos . . Cha ge a y ultiples of a gles, half a gles, etc. to e pressio s ith si gle a gles e.g., replace sin ith sin cos . c. Look for ’s. Ofte cha gi g a i to sin cos or vice versa ill e helpful. 4. Rewrite Side A i ter s of si es a d cosi es. . Factor here possi le. . Separate or co i e fractio s to The follo i g pages illustrate a u Version 2.1 ake Side A look ore like Side B. er of tech i ues that ca Page 43 of 109 e used to verify ide tities. April 10, 2017 Chapter Ide tities a d E uatio s Verifyi g Ide tities – Tech i ues Tech i ue: I vestigate O e or Both Sides Ofte , he looki g at a ide tity, it is ot i ediately o vious ho to proceed. I i vestigati g oth sides ill provide the ecessary hi ts to proceed. a y cases, Exa ple . : sin cot cot cos sin cos Yuk! This ide tity looks difficult to deal ith – there are lots of fractio s. Let’s i vestigate it y co verti g the right side to si es a d cosi es. Note that o the right, e ove the e fractio off to the side to i dicate e are i vestigati g o ly. We do this ecause e ust verify a ide tity y orki g o o ly o e side u til e get the other side. sin I cot cot cos sin cos a ipulati g the right side, e cha ged each a t so ethi g that looks cos sin cos sin cos cos cos cos i the gree e pressio to ecause e ore like the e pressio o the left. Notice that the ora ge e pressio looks a lot like the e pressio o the left, e cept that every place e have a i the e pressio o the left e have cos i the ora ge e pressio . What is our e t step? We eed to cha ge all the ’s i the e pressio o the left to cos . We ca do this y cos cos cot cot Version 2.1 ultiplyi g the e pressio o the left y ∙ sin sin , as follo s: cos cos Notice that this atches the ora ge e pressio a ove. cot cot Page 44 of 109 April 10, 2017 Chapter Ide tities a d E uatio s Verifyi g Ide tities – Tech i ues co t’d Tech i ue: Break a Fractio i to Pieces Whe a fractio co tai s ultiple ter s i the u erator, it is so eti es useful to reak it i to separate ter s. This orks especially ell he the resulti g u erator has the sa e u er of ter s as e ist o the other side of the e ual sig . Exa ple .2: cos cos cos First, it’s a good idea to replace cos cos cos sin sin cos cos tan tan ith cos cos sin sin : Ne t, reak the fractio i to t o pieces: cos cos cos cos sin sin cos cos Fi ally, si plify the e pressio : sin cos Version 2.1 ∙ sin cos tan tan tan tan Page 45 of 109 April 10, 2017 Chapter Ide tities a d E uatio s Verifyi g Ide tities – Tech i ues co t’d Tech i ue: Get a Co o De o i ator o O e Side If it looks like you ould e efit fro getti g a co o de o i ator for the t o sides of a ide tity, try co verti g o e side so that it has that de o i ator. I a y cases, this ill result i a e pressio that ill si plify i to a ore useful for . Exa ple . : cos sin sin cos If e ere to solve this like a e uatio , e ight create a co o de o i ator. Re e er, ho ever, that e ca o ly ork o o e side, so e ill o tai the co o de o i ator o o ly sin . o e side. I this e a ple, the co o de o i ator ould e: cos cos cos cos ∙ cos sin cos sin O ce e have a ipulated o e side of the ide tity to have the co o de o i ator, the rest of the e pressio should si plify. To keep the cos i the de o i ator of the e pressio o the left, e eed to ork ith the u erator. A co o su stitutio is to co vert et ee sin usi g the Pythagorea ide tity sin cos . a d cos cos sin sin Notice that the u erator is a differe ce of s uares. Let’s factor it. sin cos sin sin Fi ally, e si plify y eli i ati g the co sin cos Version 2.1 o factor i the u erator a d de o i ator. sin cos Page 46 of 109 April 10, 2017 Chapter Ide tities a d E uatio s Solvi g Trigo o etric E uatio s Solvi g trigo o etric e uatio s i volves a y of the sa e skills as solvi g e uatio s i ge eral. So e specific thi gs to atch for i solvi g trigo o etric e uatio s are the follo i g:         A u Arra ge e t. It is ofte a good idea to get arra ge the e uatio so that all ter s are o o e side of the e ual sig , a d zero is o the other. For e a ple, tan sin tan ca e rearra ged to eco e tan sin tan . Quadratics. Look for uadratic e uatio s. A y ti e a e uatio co tai s a si gle Trig fu ctio ith ultiple e po e ts, there ay e a ay to factor it like a uadratic e uatio . cos cos . For e a ple, cos Factori g. Look for ays to factor the e uatio a d solve the i dividual ter s separately. For e a ple, tan sin tan tan sin . Ter s with No Solutio . After factori g, so e ter s ill have o solutio a d ca e re uires sin , hich has o solutio si ce the discarded. For e a ple, sin si e fu ctio ever takes o a value of . Replace e t. Havi g ter s ith differe t Trig fu ctio s i the sa e e uatio is ot a pro le if you are a le to factor the e uatio so that the differe t Trig fu ctio s are i differe t factors. Whe this is ot possi le, look for ays to replace o e or ore Trig fu ctio s ith others that are also i the e uatio . The Pythagorea Ide tities are sin , particularly useful for this purpose. For e a ple, i the e uatio cos cos ca e replaced y sin , resulti g i a e uatio co tai i g o ly o e Trig fu ctio . Extra eous Solutio s. Check each solutio to ake sure it orks i the origi al e uatio . A solutio of o e factor of a e uatio ay fail as a solutio overall ecause the origi al fu ctio does ot e ist at that value. See E a ple . elo . I fi ite Nu er of Solutio s. Trigo o etric e uatio s ofte have a i fi ite u er of solutio s ecause of their periodic ature. I such cases, e appe d or a other ter to the solutio s to i dicate this. See E a ple .9 elo . Solutio s i a I terval. Be careful he solutio s are sought i a specific i terval. For the i terval , , there are typically t o solutio s for each factor co tai i g a Trig fu ctio as lo g as the varia le i the fu ctio has lead coefficie t of e.g., or θ . If the lead coefficie t is other tha e.g., or θ , the u er of solutio s ill typically e t o ultiplied y the lead coefficie t e.g., solutio s i the i terval , for a ter i volvi g . See E a ple . elo , hich has solutio s o the i terval , . er of these tech i ues are illustrated i the e a ples that follo . Version 2.1 Page 47 of 109 April 10, 2017 Chapter Ide tities a d E uatio s Solvi g Trigo o etric E uatio s – Exa ples Exa ple . : Solve for o the i terval : cos , cos The trick o this pro le is to recog ize the e pressio as a uadratic e uatio . Replace the trigo o etric fu ctio , i this case, cos , ith a varia le, like , that ill ake it easier to see ho to factor the e pressio . If you ca see ho to factor the e pressio ithout the trick, y all ea s proceed ithout it. Let cos , a d our e uatio eco es: . This e uatio factors to get: Su stituti g cos ack i for gives: cos A d fi ally: The o ly solutio for this o the i terval Exa ple . : Solve for o the i terval Whe orki g ith a pro le e pa d the i terval to , I this pro le , , , cos ⇒ is: cos √ : sin i the i terval , that i volves a fu ctio of for the first steps of the solutio . √ , so e a t all solutio s to sin here , it is useful to is a a gle i the i terval , . Note that, eyo d the t o solutio s suggested y the diagra , additio al solutio s are o tai ed y addi g ultiples of to those t o solutio s. Usi g the diagra at left, e get the follo i g solutio s: , , 7 , The , dividi g y 4, e get: Note that there are solutio s ecause the usual u er of solutio s i.e., is i creased y a factor of . Version 2.1 , 7 A d si plifyi g, e get: , , , , , , , , Page 48 of 109 , , , , , 9 , 9 , , , April 10, 2017 Chapter Ide tities a d E uatio s Solvi g Trigo o etric E uatio s – Exa ples Exa ple . : Solve for tan sin tan tan o the i terval tan sin or sin cos or , sin , , Exa ple . : Solve for cos cos cos cos cos cos ∙ cos Version 2.1 so o the i terval cos cos sin sin Exa ple .7: Solve for , is a solutio to the e uatio , tan is u defi ed at , is ot a solutio to this e uatio . , : cos cos sin , : cos cos , cos cos sin sin tan sin o the i terval cos sin While sin ,π , Use: : tan , sin sin cos cos cos cos cos sin sin sin sin ⇒ Page 49 of 109 April 10, 2017 Chapter Ide tities a d E uatio s Solvi g Trigo o etric E uatio s – Exa ples Exa ple .9: Solve for all solutio s of : sin √ sin √ sin √ The dra i g at left illustrates the t o a gles i , tan sec tan tan sec √ . To get all solutio s, e eed to add all i teger ultiples of to these solutio s. So, ∈ Exa ple . for hich sin : Solve for all solutio s of : tan sec tan or sec ∪ tan sec sec cos Collecti g the various solutio s, ∈ ∪ or ∪ Note: the solutio i volvi g the ta ge t fu ctio has t o a s ers i the i terval , . Ho ever, they are radia s apart, as ost solutio s i volvi g the ta ge t fu ctio are. Therefore, e ca si plify the a s ers y sho i g o ly o e ase a s er a d addi g , i stead of sho i g t o ase a s ers that are apart, a d addi g to each. For e a ple, the follo i g t o solutio s for tan give a ove: …, …, Version 2.1 , , , , , , , , ,… … Page 50 of 109 are telescoped i to the si gle solutio …, , , , , ,… April 10, 2017 Chapter Solvi g a O li ue Tria gle Solvi g a O li ue Tria gle Several ethods e ist to solve a o li ue tria gle, i.e., a tria gle ith o right a gle. The appropriate ethod depe ds o the i for atio availa le for the tria gle. All ethods re uire that the le gth of at least o e side e provided. I additio , o e or t o a gle easures ay e provided. Note that if t o a gle easures are provided, the easure of the third is deter i ed ecause the su of all three a gle easures ust e ˚ . The ethods used for each situatio are su arized elo . Give Three Sides a d o A gles SSS Give three seg e t le gths a d o a gle    easures, do the follo i g: Use the La of Cosi es to deter i e the easure of o e a gle. Use the La of Si es to deter i e the easure of o e of the t o re ai i g a gles. Su tract the su of the easures of the t o k o a gles fro ˚ to o tai the easure of the re ai i g a gle. Give Two Sides a d the A gle etwee The Give t o seg e t le gths a d the    SAS easure of the a gle that is et ee the , do the follo i g: Use the La of Cosi es to deter i e the le gth of the re ai i g leg. Use the La of Si es to deter i e the easure of o e of the t o re ai i g a gles. Su tract the su of the easures of the t o k o a gles fro ˚ to o tai the easure of the re ai i g a gle. Give O e Side a d Two A gles ASA or AAS Give o e seg e t le gth a d the   easures of t o a gles, do the follo i g: Su tract the su of the easures of the t o k o a gles fro ˚ to o tai the of the re ai i g a gle. Use the La of Si es to deter i e the le gths of the t o re ai i g legs. Give Two Sides a d a A gle ot etwee The easure SSA This is the A iguous Case. Several possi ilities e ist, depe di g o the le gths of the sides a d the easure of the a gle. The possi ilities are discussed o the e t several pages. Version 2.1 Page 51 of 109 April 10, 2017 Chapter Solvi g a O li ue Tria gle Laws of Si es a d Cosi es A c B b C a The tria gle a ove ca e orie ted i a y a er. It does ot Ho ever,  Side is al ays opposite across fro ∠ .  Side is al ays opposite across fro ∠ .  Side is al ays opposite across fro ∠ . atter hich a gle is , or . Law of Si es see a ove illustratio Law of Cosi es see a ove illustratio cos cos cos The la of cosi es ca e descri ed i ords as follo : The s uare of a y side is the su of the s uares of the other t o sides i us t ice the product of those t o sides a d the cosi e of the a gle et ee the . It looks a lot like the Pythagorea Theore , ith the Version 2.1 Page 52 of 109 i us ter appe ded. April 10, 2017 Chapter Solvi g a O li ue Tria gle Laws of Si es a d Cosi es – Exa ples Exa ple . : Solve the tria gle, give : A °, B . . °, a To solve: fi d the third a gle, a d the use the La of Si es. ∠ ° ° ° ° The use the La of Si es to fi d the le gths of the t o re ai i g sides. sin sin . . ° sin ° sin . ∙ sin ° sin ° ⇒ ° . ∙ sin sin ° ⇒ ° Exa ple .2: Solve the tria gle, give : a , c . ° . , B °. First, dra the tria gle fro the i for atio you are give . This ill help you get a idea of hether the values you calculate i this pro le are reaso a le. rd Ne t, fi d the le gth of the La of Cosi es: √ . 7 side of the tria gle usi g the cos cos . 7 ~ sin . sin Use the La of Si es to fi d the sin ∠ ⇒ sin sin . ° ° . . 7 easure of o e of the re ai i g a gles. 7 ° ⇒ sin . The easure of the re ai i g a gle ca e calculated either fro k o ledge that the su of the three a gles i side a tria gle is ∠ Version 2.1 ° ° ° ° Page 53 of 109 the La of Si es or fro °. April 10, 2017 Chapter Solvi g a O li ue Tria gle The A iguous Case SSA Give t o seg e t le gths a d a a gle that is ot et ee the , it is ot clear hether a tria gle is defi ed. It is possi le that the give i for atio ill defi e a si gle tria gle, t o tria gles, or eve o tria gle. Because there are ultiple possi ilities i this situatio , it is called the a iguous case. Here are the possi ilities: There are three cases i which Case : . Produces o tria gle ecause is ot lo g e ough to reach the ase. Case 2: Produces o e right tria gle ecause is e actly lo g e ough to reach the ase. for s a right a gle ith the ase, a d is the height of the tria gle. Produces t o tria gles ecause is the right size to reach the ase i t o Case : places. The a gle fro hich s i gs fro its ape to eet the ase ca take t o values. There is o e case i which Case : locatio . Version 2.1 . Produces o e tria gle ecause is too lo g to reach the ase i Page 54 of 109 ore tha o e April 10, 2017 Chapter Solvi g a O li ue Tria gle The A Solvi g the A iguous Case SSA iguous Case Ho do you solve a tria gle or t o i the a iguous case? Assu e the i for atio give is the le gths of sides a d , a d the easure of A gle . Use the follo i g steps: Step : Calculate the height of the tria gle i this develop e t, Step 2: Co pare . to the height of the tria gle, :  If , the  If , the 9 °, a d e have Case – a right tria gle. Proceed to Step 4.  If , the e have Case or Case 4. Proceed to the Step to deter i e hich. Step : Co pare   e have Case 1 – there is o tria gle. Stop here. to . , the e have Case – t o tria gles. Calculate usi g the La of Si es. Fi d If the t o a gles i the i terval °, ° ith this si e value; each of these ∠ ’s produces a separate tria gle. Proceed to Step 4 a d calculate the re ai i g values for each. , the If Step 4. Version 2.1 e have Case 4 – o e tria gle. Fi d Page 55 of 109 ∠ usi g the La of Si es. Proceed to April 10, 2017 Chapter Solvi g a O li ue Tria gle The A Solvi g the A iguous Case SSA iguous Case – co t’d Step : Calculate . At this poi t, e have the le gths of sides a d , a d the easures of A gles a d . If e are deali g ith Case – t o tria gles, e ust perfor Steps 4 a d for each tria gle. Step 4 is to calculate the easure of A gle as follo s: ∠ Step : Calculate . Fi ally, e calculate the value of usi g the La of Si es. sin sin Note: usi g A ⇒ a d∠ sin sin ay produce or sin sin ⇒ ° ∠ ∠ sin sin ore accurate results si ce oth of these values are give . iguous Case Flowchart Start Here Compare Compare to to Two triangles Calculate , and then steps and , above). Version 2.1 Page 56 of 109 April 10, 2017 Chapter Solvi g a O li ue Tria gle A iguous Case – Exa ples Exa ple . : Deter i e hether the follo i g easure e ts produce o e tria gle, t o tria gles, or o tria gle: ∠ °, a .7, c . . Solve a y tria gles that result. Si ce e are give t o sides a d a a gle that is ot et ee the , this is the a We dra this situatio ith ∠ o the left a d ha gi g do Step : Calculate . Step 2: Co pare to . .7 ∙ sin . Step : Co pare to . sin sin ∠ sin ⇒ . .7 sin °, sin or ° ∠ ° Step : sin 7. ∠ . sin Version 2.1 ⇒ ⇒ sin . ° ° ° Tria gle 2 – Start with: .7, ∠ Step : ° sin ° ust solve each. ° ° . ∠ Tria gle – Start with: Step : . ° have this si e value. Let’s fi d the : Si ce e ill have t o tria gles, e °, .7 .7, so e have Case – t o tria gles. . T o a gles i the i terval ∠ .7 elo . usi g the La of Si es: Calculate sin .7, ° , as sho iguous case. ∠ ° ° sin °, ° Step : . ° sin Page 57 of 109 . ∠ . sin ° ° ⇒ ° sin 7° 7° sin . ° April 10, 2017 Chapter Solvi g a O li ue Tria gle A iguous Case – Exa ples Exa ple . : Deter i e hether the follo i g easure e ts produce o e tria gle, t o tria gles, or o tria gle: ∠B °, b , a . Solve a y tria gles that result. Si ce e are give t o sides a d a a gle that is ot et ee the , this is the a We dra this situatio Step : Calculate Step 2: Co pare ith ∠ o the left a d . ∙ sin to . .9 ha gi g do ° , as sho iguous case. elo . .9 . Stop. We have Case 1 – o tria gle. Alter ative Method Calculate the sin ∠ sin sin easure of a gle ⇒ . 9 sin sin usi g the La of Si es: ° ⇒ sin . 9 . 9 is ot a valid si e value recall that si e values ra ge fro values do ot defi e a tria gle. Note: The Alter ative Method for deali g ith the a Appe di B. Version 2.1 Page 58 of 109 to . Therefore, the give iguous case is laid out i detail i April 10, 2017 Chapter Solvi g a O li ue Tria gle Beari gs Beari gs are descri ed differe tly fro other a gles i Trigo o etry. A eari g is a clock ise or cou terclock ise a gle hose i itial side is either due orth or due south. The stude t ill eed to tra slate these i to refere ce a gles a d/or polar a gles to solve pro le s i volvi g eari gs. So e eari gs, alo g ith the key associated a gles are sho i the illustratio s elo . The eari g a gle is sho as , the refere ce a gle is sho as , a d the polar a gle is sho as . Beari g: Beari g A gle: β ° Refere ce A gle: θ Polar A gle: Beari g: Beari g A gle: β ° Refere ce A gle: θ Polar A gle: Version 2.1 Beari g: ° ° Beari g A gle: β Refere ce A gle: θ ° Polar A gle: Beari g: ° ° ° Beari g A gle: β ° Refere ce A gle: θ ° Polar A gle: Page 59 of 109 ° ° 7 ° ° ° ° April 10, 2017 Chapter Solvi g a O li ue Tria gle Beari gs – Exa ples Exa ple . : T o tracki g statio s are o the e uator 1 7 iles apart. A eather alloo is located o a eari g of N ° E fro the ester statio a d o a eari g of N ° W fro the easter statio . Ho far is the alloo fro the ester statio ? The eari g a gles give are those sho i ora ge i the diagra at right. The first step is to calculate the refere ce a gles sho i age ta i the diagra . 9 ° ° θ ° 9 ° ° ° 77° ° 77° 9° The , use the La of Si es, as follo s: 7 sin 9° sin 77° ⇒ . miles Exa ple . : T o sail oats leave a har or i the Baha as at the sa e ti e. The first sails at ph i a directio S ° E. The seco d sails at ph i a directio S 7 ° W. Assu i g that oth oats ai tai speed a d headi g, after 4 hours, ho far apart are the oats? Let’s dra a diagra to illustrate this situatio . The le gths of t o sides of a tria gle are ased o the dista ces the oats travel i four hours. The eari g a gles give are used to calculate the refere ce sho i ora ge i the diagra elo . Boat 1 travels: mph ∙ hours miles at a headi g of S ° E. This gives a refere ce a gle of 9 ° ° ° elo the positive ‐a is. Boat travels: mph ∙ hours mi. at a headi g of S 7 ° W. This gives a refere ce a gle of 9 ° 7 ° ° elo the egative ‐a is. Usi g the La of Cosi es, e ca calculate: cos Version 2.1 ° Page 60 of 109 , ⇒ 9 . miles April 10, 2017 Chapter 7 Area of a Tria gle Area of a Tria gle Area of a Tria gle There are a u er of for ulas for the area of a tria gle, depe di g o a out the tria gle is availa le. hat i for atio Geo etry For ula: This for ula, lear ed i Ele e tary Geo etry, is pro a ly ost fa iliar to the stude t. It ca e used he the ase a d height of a tria gle are either k o or ca e deter i ed. here, is the le gth of the ase of the tria gle. is the height of the tria gle. Note: The ase ca e a y side of the tria gle. The height is the le gth of the altitude of hichever side is selected as the ase. So, you ca use: or or Hero ’s For ula: Hero ’s for ula for the area of a tria gle ca e used he the le gths of all of the sides are k o . So eti es this for ula, though less appeali g, ca e very useful. . here, Note: , , are the le gths of the sides of the tria gle. is called the se i‐peri eter of the tria gle ecause it is half of the tria gle’s peri eter. Version 2.1 Page 61 of 109 April 10, 2017 Chapter 7 Area of a Tria gle Area of a Tria gle co t’d Trigo o etric For ulas The follo i g for ulas for the area of a tria gle ca e derived fro the Geo etry for ula, , usi g Trigo o etry. Which o e to use depe ds o the i for atio availa le: Two a gles a d o e side: ∙ ∙ ∙ ∙ Two sides a d the a gle etwee the : ∙ ∙ ∙ ∙ ∙ Coordi ate Geo etry For ula If the three vertices of a tria gle are displayed i a coordi ate pla e, the for ula elo , usi g a deter i a t, ill give the area of a tria gle. Let vertices of a tria gle i the coordi ate pla e e: area of the tria gle is: , , , , , . The , the ∙ Exa ple 7. : For the tria gle i the figure at right, the area is: ∙ ∙ ∙| Version 2.1 | ∙ 7 Page 62 of 109 7 April 10, 2017 Chapter 7 Area of a Tria gle Area of a Tria gle – Exa ples Exa ple 7.2: Fi d the area of the tria gle if: C = 1 sin ∙ ∙ ∙ sin ° ∙ . °, a = 4 yards, = yards. yards Exa ple 7. : Fi d the area of the tria gle if: yards, yards, yards. To solve this pro le , e ill use Hero ’s for ula: First calculate: The , √ ∙ ∙7∙ .99 yards √ Exa ple 7. : Fi d the area of the tria gle i the figure elo usi g Coordi ate Geo etry: ∙ ∙ ∙ ∙| 7 7 9 | 7 ∙ Note: It is easy to see that this tria gle has a ase of le gth Ele e tary Geo etry, the area of the tria gle is: a d a height of , so fro ∙ ∙ sa e a s er . The stude t ay ish to test the other ethods for calculati g area that are prese ted i this chapter to see if they produce the sa e result. Hi t: they do. Version 2.1 Page 63 of 109 April 10, 2017 Chapter Polar Coordi ates Polar Coordi ates Polar coordi ates are a alter ative ethod of descri i g a poi t i a Cartesia pla e ased o the dista ce of the poi t fro the origi a d the polar a gle hose ter i al side co tai s the poi t. Let’s take a look at the relatio ship et ee a poi t’s recta gular coordi ates coordi ates , . The ag itude, r, is the dista ce of the poi t fro a d its polar , the origi : The a gle, θ, is the polar a gle hose ter i al side co tai s the poi t. Ge erally, this a gle is e pressed i radia s, ot degrees: tan Co versio fro cos so , adjusted to e i the appropriate uadra t. tan polar coordi ates to recta gular coordi ates is straightfor ard: a d sin Exa ple . : E press the recta gular for coordi ates: Give : so tan tan tan , √ i polar i Quadra t II, So, the coordi ates of the poi t are as follo s: Recta gular coordi ates: Exa ple .2: E press the polar for Give : cos sin √ √ ∙ cos √ ∙ sin Polar Coordi ates: , i recta gular coordi ates: √ , √ ∙ √ ∙ √ , √ √ So, the coordi ates of the poi t are as follo s: Polar Coordi ates: Version 2.1 Recta gular coordi ates: √ , Page 64 of 109 , April 10, 2017 Chapter Polar Coordi ates Polar For Expressi g Co plex Nu of Co plex Nu ers ers i Polar For A co ple u er ca e represe ted as poi t i the Cartesia Pla e, usi g the horizo tal a is for the real co po e t of the u er a d the vertical a is for the i agi ary co po e t of the u er. If e e press a co ple polar coordi ates as t o for s for are: u er i recta gular coordi ates as , e ca also e press it i cos sin , ith ∈ , . The , the e uivale ces et ee the Co vert Recta gular to Polar Mag itude: | | A gle: Co vert Polar to Recta gular ‐coordi ate: √ tan y‐coordi ate: Si ce ill ge erally have t o values o resides. uadra t i hich Operatio s o Co plex Nu cos sin , e eed to e careful to select the a gle i the , ers i Polar For Arou d 174 , Leo hard Euler proved that: cos co ple u er as a e po e tial for of . That is: sin . As a result, e ca e press a y cos sin ∙ ∙ , the follo i g rules regardi g Thi ki g of each co ple u er as ei g i the for operatio s o co ple u ers ca e easily derived ased o the properties of e po e ts. Let: cos Multiplicatio : So, to sin ∙ ultiply co ple cos u u Po ers: This results directly fro Roots: √ √ This results directly fro Version 2.1 sin ers, you Divisio : So, to divide co ple , ultiply their cos sin ers, you divide their cos the cos sin . The , ag itudes a d add their a gles. ag itudes a d su tract their a gles. sin ultiplicatio rule. cos sin also, see DeMoi re’s Theorem belo the po er rule if the e po e t is a fractio . Page 65 of 109 April 10, 2017 Chapter Polar Coordi ates Operatio s o Co plex Nu Exa ple . : Fi d the product: cos √ cos √ To ultiply t o u shortha d is: shortha d is: sin ers i polar for , √ ∙ √ ∙ cis ∙ . ∙ sin ers ‐ Exa ples √ cis √ cis ∙ √ ∙ √ ultiply the ‐values a d add the a gles. 9 7 √ cis √ cis √ because cis . Note: ultiplicatio ay e easier to u dersta d i e po e tial for , si ce e po e ts are added he values ith the sa e ase are ultiplied: ∙ √ ∙ ∙ √ √ ∙√ ∙ Exa ple . : Fi d the uotie t: cos √ To divide t o u √ ∙ √ √ ∙ cis cis √ shortha d is: √ cis √ . sin cos √ √ shortha d is: sin √ cis ∙ √ ∙ √ ers i polar for , divide the ‐values a d su tract the a gles. 9 7 √ √ cis cis √ because cis i. Note: divisio ay e easier to u dersta d i e po e tial for , si ce e po e ts are su tracted he values ith the sa e ase are divided: √ √ Version 2.1 ∙ ∙ √ √ √ Page 66 of 109 √ √ April 10, 2017 Chapter Polar Coordi ates DeMoivre’s Theore A raha de Moivre 1 7‐17 4 as a Fre ch athe aticia for deali g ith operatio s o co ple u ers. If e let page: cos , DeMoivre’s Theore sin Exa ple . : Fi d First, si ce , e have The , √7 A d, tan So, √ . ° ~ , Exa ple . : Fi d √ First, si ce The , A d, tan , So, √ Version 2.1 ; . 9 ° i Q . √ 9. . . ° , 9.9 , e have . ° ~ 9. , 9 ∙ cos √ a d . ° sin . ° . ° i Q ° cis 9. . , 9 √7. ° ; √ sin a d , 9 cis √7 gives us the po er rule e pressed o the prior cos √7 ho developed a very useful Theore . ° ∙ cos 9. Page 67 of 109 ° sin 9. ° April 10, 2017 Chapter Polar Coordi ates DeMoivre’s Theore Let cos e uidista t fro these roots ca sin . The , has disti ct co ple ‐th roots that occupy positio s each other o a circle of radius √ . Let’s call the roots: e calculated as follo s , , ,… , : sin √ ∙ cos The for ula could also e restated ith First, si ce The , √ tan . a d ; √ √ °; The i cre e tal a gle for successive roots is: ~ . 9 . ° Angle . ° .7 ° 7 ° .7 ° ° 7 ° .7 ° 7 ° ° ° 7 ° .7 .7 . .7 7 .7 ° i Q4 √ √ . The , ,… , . The create a chart like this: Fifth roots of , ° if this helps i the calculatio . . , e have , √ ∙ cis replaced y Exa ple .7: Fi d the fifth roots of A d, for Roots ° ° roots ~ . √ ∙ 7 °. . . . 7 . 77 7 . 7 . √ ∙ . . ° ∙ 7 .9 9 . . Notice that if e add a other 7 °, e get .7 °, hich is e uivale t to our first a gle, . ° ecause .7 ° ° . °. This is a good thi g to check. The e t a gle ill al ays e e uivale t to the first a gle! If it is ’t, go ack a d check your ork. Roots fit o a circle: Notice that, si ce all of the roots of have the sa e ag itude, a d their a gles are 7 ° apart fro each other, they occupy e uidista t positio s o a circle ith ce ter , a d radius √ Version 2.1 √ ~ . 9 . Page 68 of 109 April 10, 2017 Chapter 9 Polar Fu ctio s Polar Graphs Typically, Polar Graphs ill e plotted o polar graph paper such as that illustrated at right. O this graph, a poi t , ca e co sidered to e the i tersectio of the circle of radius a d the ter i al side of the a gle see the illustratio elo . Note: a free PC app that ca e used to desig a d pri t your o polar graph paper is availa le at . athguy.us. Parts of the Polar Graph The illustratio elo sho s the key parts of a polar graph, alo g ith a poi t, The Pole is the poi t , , . i.e., the origi . The Polar A is is the positive ‐a is. The Li e: is the positive ‐a is. Ma y e uatio s that co tai the cosi e fu ctio are sy etric a out the ‐a is. Ma y e uatio s that co tai the si e fu ctio are sy etric a out the ‐a is. Polar E uatio s – Sy Follo i g are the three Sy etry a out: Pole ‐axis ‐axis 1 etry ai types of sy etry e hi ited i Quadra ts Co tai i g Sy a y polar e uatio graphs: etry Opposite I a d III or II a d IV Left he isphere II a d III) or right he isphere I a d IV Upper he isphere I a d II) or lo er he isphere III a d IV Sy Replace ith – i the e uatio Replace Replace e uatio etry Test , ith – i the e uatio ith , i the If perfor i g the i dicated replace e t results i a e uivale t e uatio , the e uatio passes the sy etry test a d the i dicated sy etry e ists. If the e uatio fails the sy etry test, sy etry ay or ay ot e ist. Version 2.1 Page 69 of 109 April 10, 2017 Chapter 9 Polar Fu ctio s Graphs of Polar E uatio s Graphi g Methods Method : Poi t plotti g     Create a t o‐colu chart that calculates values of for selected values of . This is aki to a t o‐colu chart that calculates values of for selected values of that ca e used to plot a recta gular coordi ates e uatio e.g., . The ‐values you select for purposes of poi t plotti g should vary depe di g o the e uatio you are orki g ith i particular, the coefficie t of i the e uatio . Ho ever, a safe et is to start ith ultiples of i cludi g . Plot each poi t o the polar graph a d see hat shape e erges. If you eed ore or fe er poi ts to see hat curve is e ergi g, adjust as you go. If you k o a ythi g a out the curve typical shape, sy etry, etc. , use it to facilitate plotti g poi ts. Co ect the poi ts ith a s ooth curve. Ad ire the result; a y of these curves are aesthetically pleasi g. Method 2: Calculator Usi g a TI‐ 4 Plus Calculator or its e uivale t, do the follo i g:     Make sure your calculator is set to radia s a d polar fu ctio s. Hit the MODE key; select RADIANS i ro 4 a d POLAR i ro . After you do this, hitti g CLEAR ill get you ack to the ai scree . Hit Y= a d e ter the e uatio i the for . Use the X,T, , key to , you ay e ter θ i to the e uatio . If your e uatio is of the for a d , a d plot oth. eed to e ter t o fu ctio s, Hit GRAPH to plot the fu ctio or fu ctio s you e tered i the previous step. If ecessary, hit WINDOW to adjust the para eters of the plot. o If you ca ot see the hole fu ctio , adjust the X‐ a d Y‐ varia les or use )OOM . o If the curve is ot s ooth, reduce the value of the step varia le. This ill plot ore poi ts o the scree . Note that s aller values of step re uire ore ti e to plot the curve, so choose a value that plots the curve ell i a reaso a le a ou t of ti e. o If the e tire curve is ot plotted, adjust the values of the i a d ax varia les u til you see hat appears to e the e tire plot. Note: You ca vie the ta le of poi ts used to graph the polar fu ctio Version 2.1 Page 70 of 109 y hitti g 2ND – TABLE. April 10, 2017 Chapter 9 Polar Fu ctio s Graph of Polar E uatio s Circle E uatio : Locatio : a ove ‐a is if ‐a is if elo Radius: Sy / etry: sin ‐a is E uatio : E uatio : cos Locatio : right of ‐a is if left of ‐a is if Locatio : Ce tered o the Pole Radius: Radius: Sy / etry: ‐a is Sy etry: Pole, ‐a is, ‐a is Rose Characteristics of roses:       sin E uatio : o Sy etric a out the ‐a is E uatio : cos o Sy etric a out the ‐a is Co tai ed ithi a circle of radius If is odd, the rose has petals. If is eve the rose has petals. Note that a circle is a rose ith o e petal i.e, Version 2.1 Page 71 of 109 . April 10, 2017 Chapter 9 Polar Fu ctio s Graphs of Polar E uatio s Li aço of Pascal E uatio : E uatio : sin Locatio : ul a ove ‐a is if ‐a is if ul elo Sy Locatio : ul right of ‐a is if ul left of ‐a is if ‐a is etry: cos Sy etry: ‐a is Four Li aço Shapes I er loop Cardioid Di ple No di ple Four Li aço Orie tatio s usi g the Cardioid as a e a ple si e fu ctio Version 2.1 si e fu ctio cosi e fu ctio Page 72 of 109 cosi e fu ctio April 10, 2017 Chapter 9 Polar Fu ctio s Graph of Polar E uatio s Le iscate of Ber oulli The le iscate is the set of all poi ts for hich the product of the dista ces fro t o poi ts i.e., foci apart is . hich are Characteristics of le    E uatio : o Sy E uatio : o Sy iscates: sin etric a out the li e cos etric a out the ‐a is Co tai ed ithi a circle of radius Spirals Hyper olic Spiral Archi edes’ Spiral Fer at’s Spiral Lituus Characteristics of spirals:   , E uatio : o Dista ce fro the Pole i creases ith E uatio : , o Hyper olic Spiral  o Lituus : asy ptotic to the li e u its fro the ‐a is : asy ptotic to the ‐a is Not co tai ed ithi a y circle Version 2.1 Page 73 of 109 April 10, 2017 Chapter 9 Polar Fu ctio s Graphi g Polar E uatio s – The Rose Exa ple 9. : This fu ctio is a rose. Co sider the for s The u sin a d er of petals o the rose depe ds o the value of .  If is a eve i teger, the rose ill have petals.  If is a odd i teger, it ill have petals. cos . Let’s create a ta le of values a d graph the e uatio : / / . / . / / / 4 7π/ π/ π/ / / . Because this fu ctio i volves a argu e t of , e a t to start y looki g at values of θ i , , . You could plot ore poi ts, ut this i terval is sufficie t to esta lish the ature of the curve; so you ca graph the rest easily. ‐4 . The values i the ta le ge erate the poi ts i the t o petals right of the ‐a is. O ce sy etry is esta lished, these values are easily deter i ed. Blue poi ts o the graph correspo d to lue values i the ta le. K o i g that the curve is a rose allo s us to graph the other t o petals ithout calculati g ore poi ts. Ora ge poi ts o the graph correspo d to ora ge values i the ta le. The four Rose for s: Version 2.1 Page 74 of 109 April 10, 2017 Chapter 9 Polar Fu ctio s Graphi g Polar E uatio s – The Cardioid Exa ple 9.2: This cardioid is also a li aço of for sin ith . The use of the si e fu ctio i dicates that the large loop ill e sy etric a out the ‐a is. The sig i dicates that the large loop ill e a ove the ‐a is. Let’s create a ta le of values a d graph the e uatio : / / .7 / .7 / / 4 7π/ π/ . / . π/ / The portio of the graph a ove the ‐a is results fro i Q1 a d Q , here the si e fu ctio is positive. Ge erally, you a t to look at values of i , . Ho ever, so e fu ctio s re uire larger i tervals. The size of the i terval depe ds largely o the ature of the fu ctio a d the coefficie t of . O ce sy etry is esta lished, these values are easily deter i ed. Blue poi ts o the graph correspo d to lue values i the ta le. Si ilarly, the portio of the graph elo the ‐a is results fro i Q a d Q4, here the si e fu ctio is egative. Ora ge poi ts o the graph correspo d to ora ge values i the ta le. The four Cardioid for s: Version 2.1 Page 75 of 109 April 10, 2017 Chapter 9 Polar Fu ctio s Co verti g Betwee Polar a d Recta gular For s of E uatio s Recta gular to Polar To co vert a e uatio fro Recta gular For cos Substitute sin Substitute Substitute Exa ple 9. : Co vert to Polar For , use the follo i g e uivale ces: cos sin for for for to a polar e uatio of the for . Starti g E uatio : Su stitute Factor out : Divide y a d cos cos sin : ∙ cos cos : sin ∙ sin sin Polar to Recta gular To co vert a e uatio fro Polar For cos Substitute sin Substitute Substitute Exa ple 9. : Co vert r = cos Starti g E uatio : Su stitute cos , sin Multiply y : Su stitute Su tract to Recta gular For , use the follo i g e uivale ces: + 9 sin : for cos for sin for to a recta gular e uatio . r = cos 9 : Co plete the s uare: Version 2.1 9 9 : Si plify to sta dard for + 9 sin for a circle: Page 76 of 109 9 9 9 April 10, 2017 Chapter 9 Polar Fu ctio s Para etric E uatio s O e ay to defi e a curve is y aki g a d or a d fu ctio s of a third varia le, ofte for ti e . The third varia le is called the Para eter, a d fu ctio s defi ed i this a er are said to e i Para etric For . The e uatio s that defi e the desired fu ctio are called Para etric E uatio s. I Para etric E uatio s, the para eter is the i depe de t varia le. Each of the other t o or ore varia les is depe de t o the value of the para eter. As the para eter cha ges, the other varia les cha ge, ge erati g the poi ts of the fu ctio . Exa ple 9. : A relatively si ple e a ple is a circle, hich e ca defi e as follo s: Circle: cos As the varia le progresses fro sin to The circle i the illustratio at right ca , a circle of radius is or . e defi ed i several ays: Cartesia for : Polar for : Para etric for : Fa iliar Curves cos Ma y curves ith hich the stude t follo i g: Curve sin ay e fa iliar have para etric for s. A o g those are the Cartesia For Para ola ith horizo tal directri Polar For sin Ellipse ith horizo tal ajor a is ∙ cos Hyper ola ith horizo tal tra sverse a is ∙ cos Para etric For cos sin sec tan As ca e see fro this chart, so eti es the para etric for of a fu ctio is its si plest. I fact, para etric e uatio s ofte allo us to graph curves that ould e very difficult to graph i either Polar for or Cartesia for . So e of these are illustrated o the e t page. Version 2.1 Page 77 of 109 April 10, 2017 Chapter 9 Polar Fu ctio s So e Fu ctio s Defi ed y Para etric E uatio s Star Wars fa s: are these the oids you are looki g for? The graphs elo are e a ples of fu ctio s defi ed y para etric e uatio s. The e uatio s a d a rief descriptio of the curve are provided for each fu ctio . Deltoid Nephroid Para etric e uatio s: cos sin cos sin The deltoid is the path of a poi t o the circu fere ce of a circle as it akes three co plete revolutio s o the i side of a larger circle. Astroid Para etric e uatio s: cos sin cos sin The ephroid is the path of a poi t o the circu fere ce of a circle as it akes t o co plete revolutio s o the outside of a larger circle. Para etric e uatio s: cos sin The astroid is the path of a poi t o the circu fere ce of a circle as it akes four co plete revolutio s o the i side of a larger circle. Cycloid Para etric e uatio s: sin cos Version 2.1 The cycloid is the path of a poi t o the circu fere ce of a circle as the circle rolls alo g a flat surface thi k: the path of a poi t o the outside of a icycle tire as you ride o the side alk . The cycloid is oth a brachistochrone a d a tautochrone look these up if you are i terested . Page 78 of 109 April 10, 2017 Chapter 1 Vectors Vectors A vector is a ua tity that has oth ag itude a d directio . A e a ple ould e i d lo i g to ard the east at iles per hour. A other e a ple ould e the force of a 1 kg eight ei g pulled to ard the earth a force you ca feel if you are holdi g the eight . Special U it Vectors We defi e u it vectors to e vectors of le gth . U it vectors havi g the directio of the positive a es are very useful. They are descri ed i the chart a d graphic elo . U it Vector Directio Graphical represe tatio of positive ‐a is u it vectors a d j i t o di e sio s. positive ‐a is positive ‐a is Vector Co po e ts The le gth of a vector, , is called its ag itude a d is represe ted y the sy ol ‖ ‖. If a , , , a d its ter i al poi t e di g positio is vector’s i itial poi t starti g positio is , , , the the vector displaces i the ‐directio , i the ‐ directio , a d i the ‐directio . We ca , the , represe t the vector as follo s: The ag itude of the vector, , is calculated as: ‖ ‖ √ If this looks fa iliar, it should. The ag itude of a vector i three di es sio s is deter i ed as the le gth of the space diago al of a recta gular pris ith sides , a d . I t o di e sio s, these co cepts co tract to the follo i g: ‖ ‖ √ I t o di e sio s, the ag itude of the vector is the le gth of the hypote use of a right tria gle ith sides a d . Version 2.1 Page 79 of 109 April 10, 2017 Chapter 1 Vectors Vector Properties Vectors have a u er of ice properties that ake orki g ith the oth useful a d relatively si ple. Let a d e scalars, a d let u, v a d w e vectors. The ,      If , the ‖ ‖ cos a d ote: this for ula is ofte ‖ ‖ cos ‖ ‖ sin The , used i Force calculatio s a d If If ‖ ‖ sin , the , the Defi e to e the zero vector i.e., it has zero le gth, so that zero vector is also called the ull vector. . Note: the Note: ca also e sho ith the follo i g otatio : , useful i calculati g dot products a d perfor i g operatio s ith vectors. . This otatio is Properties of Vectors  Additive Ide tity  Additive I verse  Co  utative Property Associative Property  Associative Property  Distri utive Property  Distri utive Property  Multiplicative Ide tity Also, ote that:   ‖ ‖ ‖ Version 2.1 ‖ | |‖ ‖ Mag itude Property U it vector i the directio of Page 80 of 109 April 10, 2017 Chapter 1 Vectors Vector Properties – Exa ples Exa ple . : u = ‐ i ‐ j, v = i + j; Fi d u + v. A alter ative otatio for a vector i the for is , . Usi g this alter ative otatio akes a y vector operatio s uch easier to ork ith. To add vectors, si ply li e the vertically a d add: up , Exa ple .2: u = ‐ i ‐ 7j a d v = ‐4i ‐ 1j; Fi d ‖ , , ‖ Exa ple ‖ √ 7 , √ ‖. Su tracti g , , , is the sa e as addi g . To get – , si ply cha ge the sig of each ele e t of . If you fi d it easier to add tha to su tract, you ay a t to adopt this approach to su tracti g vectors. ∙√ √ . : Fi d the u it vector that has the sa e directio as the vector v = i ‐ 1 j. A u it vector has ag itude . To get a u it vector i the sa e directio as the origi al vector, divide the vector y its ag itude. The u it vector is: Version 2.1 ‖ ‖ √ Page 81 of 109 April 10, 2017 Chapter 1 Vectors Vector Properties – Exa ples Exa ple . : Write the vector v i ter s of i a d j if ‖ ‖ = 1 a d directio a gle θ = 1 °. It helps to graph the vector ide tified i the pro le . The u it vector i the directio θ cos °, sin Multiply this y ‖ ‖ Version 2.1 ° √ , ° is: √ to get : √ √ Page 82 of 109 April 10, 2017 Chapter 1 Vectors Vector Dot Product The Dot Product of t o vectors, follo s: ∘ a d ∙ , is defi ed as ∙ ∙ It is i porta t to ote that the dot product is a scalar i.e., a u er , ot a vector. It descri es so ethi g a out the relatio ship et ee t o vectors, ut is ot a vector itself. A useful approach to calculati g the dot product of t o vectors is illustrated here: , alter ative vector otatio , , , Ge eral I the e a ple at right the vectors are li ed up vertically. The u ers i the each colu are ultiplied a d the results are added to get the dot product. I the e a ple, , , ∘ , , . ∘ , , , , Exa ple ∘ , , , , Properties of the Dot Product Let  e a scalar, a d let u, v a d w e vectors. The , ∘  ∘  ∘ ∘  ∘ ‖ ‖ ∘  ∘  ∘ )ero Property   If ∘ ∘ ∘ ∘ Version 2.1 ∘ a d If there is a scalar If Co are orthogo al to each other. utative Property Mag itude S uare Property More properties:  , a d ∘ ∘ a d such that is the a gle et ee Distri utive Property Multiplicatio , the y a Scalar Property a d are orthogo al perpe dicular . , the a d , the cos Page 83 of 109 a d are parallel. ∘ ‖ ‖‖ ‖ . April 10, 2017 Chapter 1 Vectors Vector Dot Product – Exa ples Exa ple . : u = ‐ i + j, v = i ‐ j, w = ‐ i + 1 j; Fi d u ∙ w + v ∙ w. The alter ate otatio for vectors co es i especially ha dy i doi g these types of pro le s. Also, ote that: u ∙ w + v ∙ w = u + v ∘ w. Let’s calculate u + v ∘ w. u Exa ple , , ∘ v ∘w ∙ Usi g the distri utive property for dot products results i a easier pro le ith fe er calculatio s. , , ∙ . : Fi d the a gle et ee the give vectors: u = i ‐ j, v = 4i + j. ∘ cos ‖ ‖ ‖ ‖ ° , , ∘ ∘ ° ‖ ‖ ‖ ‖ cos Exa ple ∙ ∙ ∘ cos ‖ ‖ ‖ ‖ √ √ √ √ ∙√ √ 9 . ° .7: Are the follo i g vectors parallel, orthogo al, or either? v = 4i + j, w = i ‐ 4j If vectors are parallel, o e is a ultiple of the other; also If vectors are perpe dicular, their dot product is zero. ∘ ‖ ‖ ‖ ‖. Calculate the dot product. ∘ ∘ Version 2.1 , , ∙ So, the vectors are orthogo al. ∙ Page 84 of 109 April 10, 2017 Chapter 1 Vectors Vector Dot Product – Exa ples Exa ple . : Are the vectors are parallel, orthogo al, or either. v = i + 4j, w = i + j Vector Multiple Approach Clearly, It is clearly easier to check hether o e vector is a ultiple of the other tha to use the dot product ethod. The stude t ay use either, u less i structed to use a particular ethod. , , The vectors are parallel. Dot Product Approach To deter i e if t o vectors are parallel usi g the dot product, e check to see if: ∘ ∘ ∘ ‖ ‖ ‖ ‖ , , ‖ ‖ ‖ ‖ ‖ ‖ ‖ ‖ ∙ ∘ The vectors are parallel. Cross Product Approach see Cross Product elo To deter i e if t o vectors are parallel usi g the cross product, e check to see if: x x x v w v w The vectors are parallel. Version 2.1 ∙ v w ∙ v w Page 85 of 109 April 10, 2017 Chapter 1 Vectors Applicatio s of the Vector Dot Product Vector Projectio The projectio of a vector, , o to a other vector , is o tai ed usi g the dot product. The for ula used to deter i e the projectio vector is: Notice that I the diagra proj ∘ ‖ ‖ ∘ ∘ ‖ ‖ ∘ is a scalar, a d that proj at right, v proj is a vector. . Orthogo al Co po e ts of a Vector Deco positio A vector, , ca e e pressed as the su of t o orthogo al vectors a ove diagra . The resulti g vectors are: ∘ proj is parallel to a d , as sho i the a d ‖ ‖ is orthogo al to Work Work is a scalar ua tity i physics that easures the force e erted o a o ject over a particular dista ce. It is defi ed usi g vectors, as sho elo . Let:   F e the force vector acti g o a o ject, e the vector fro  The , e defi e ork as: ‖ ‖ Mag itude of Force Version 2.1 poi t to poi t . to . e the a gle et ee F a d ∘ ovi g it fro cos Dista ce Traveled . Both of these for ulas are useful. Which o e to use i a particular situatio depe ds o hat i for atio is availa le. A gle et ee Vectors Page 86 of 109 April 10, 2017 Chapter 1 Vectors Applicatio s of Vectors – Exa ples Exa ple .9: The ag itude a d directio of t o forces acti g o a o ject are pou ds, N ° E, a d 7 pou ds, N ° W, respectively. Fi d the ag itude a d the directio a gle of the resulta t force. This pro le re uires the additio of t o vectors. The approach used here is: 1 Co vert each vector i to its i a d j co po e ts, call the a d , Add the resulti g a d values for the t o vectors, a d Co vert the su to its polar for . Keep additio al accuracy throughout a d rou d at the e d. This ill preve t error co pou di g a d ill preserve the re uired accuracy of your fi al solutio s. Step 1: Co vert each vector i to its i a d j co po e ts Let Fro θ Let Fro φ e a force of the diagra 9 ° cos sin l s. at eari g: N at right, ° ° ° . ° 7 .9 7 e a force of 7 l s. at eari g: N the diagra 9 ° 7 cos 7 sin at right, ° ° ° ° °E °W .99 . Step : Add the results for the t o vectors . 7 , .99 , . 7 , 99. Step : Co vert the su Directio A gle Mag itude Version 2.1 .9 . θ 7 to its polar for tan . . 7 . 99. 9 .7° Page 87 of 109 99.79 l s. April 10, 2017 Chapter 1 Vectors Applicatio s of Vectors – Exa ples Exa ple . : O e rope pulls a arge directly east ith a force of 79 e to s, a d a other rope pulls the arge directly orth ith a force of 7 e to s. Fi d the ag itude a d directio a gle of the resulti g force acti g o the arge. The process of addi g t o vectors hose headi gs are orth, east, est or south NEWS is very si ilar to co verti g a set of recta gular coordi ates to polar coordi ates. So, if this process see s fa iliar, that’s ecause it is. Mag itude Directio A gle 79 θ tan 7. 7 7. ° newtons Exa ple . : A force is give y the vector F = i + j. The force oves a o ject alo g a straight li e fro the poi t , 7 to the poi t 1 , 1 . Fi d the ork do e if the dista ce is easured i feet a d the force is easured i pou ds. For this pro le it is sufficie t to use the ork for ula, We are give We ca calculate , . ∙ as the differe ce et ee the t o give poi ts. , , 7 The , calculate ∘ ∘ Version 2.1 , ∘ Note that the differe ce et ee t o poi ts is a vector. , ∙ , ∙ 77 foot pou ds Page 88 of 109 April 10, 2017 Chapter 1 Vectors Applicatio s of Vectors – Exa ples Exa ple . 2: Deco pose i to t o vectors orthogo al to . = i ‐ 4j, = i + j a d , here is parallel to w a d is The for ulas for this are: ∘ proj ‖ ‖ Let’s do the calculatio s. ∘ ∘ ‖ ‖ , , ∙ ∙ The , ∘ proj ‖ ‖ , , A d, , 9 Version 2.1 , , Page 89 of 109 April 10, 2017 Chapter 1 Vectors Vector Cross Product Cross Product I three di e sio s, Let: u u a d u The , the Cross Product is give u v x u v y: u v u v ‖ ‖ ‖ ‖ sin x v v u v v u v u v u v u v Expla atio : The cross product of t o o zero vectors i three di e sio s produces a third vector that is orthogo al to each of the first t o. This resulti g vector x is, therefore, or al to the pla e co tai i g the first t o vectors assu i g a d are ot parallel . I the seco d for ula a ove, is the u it vector or al to the pla e co tai i g the first t o vectors. Its orie tatio directio is deter i ed usi g the right ha d rule. Right Ha d Rule x Usi g your right ha d:  Poi t your forefi ger i the directio of , a d  Poi t your iddle fi ger i the directio of . The :  Your thu ill poi t i the directio of x . I t o di e sio s, Let: The , u x u u v a d u v v u v v u v hich is a scalar i t o di e sio s . The cross product of t o o zero vectors i t o di e sio s is zero if the vectors are parallel. That is, vectors a d are parallel if x . The area of a parallelogra ‖ ‖ ‖ ‖ sin θ. Version 2.1 havi g a d as adjace t sides a d a gle θ et ee the : Page 90 of 109 April 10, 2017 Chapter 1 Vectors Vector Cross Product Properties of the Cross Product Let  e a scalar, a d let u, v a d w e vectors. The , x  x   x  x , x   , )ero Property x m x x , x x , , a d are orthogo al to each other Reverse orie tatio orthogo ality x Every o ‐zero vector is parallel to itself A ti‐co x x  x x Distri utive Property x x x x m m utative Property Distri utive Property Scalar Multiplicatio x More properties:  If  If x , the a d are parallel. is the a gle et ee ‖ a d , the sin ‖ ‖ ‖‖ ‖ . A gle Betwee Two Vectors Notice the si ilarities i the for ulas for the a gle et ee t o vectors usi g the dot product a d the cross product: cos Version 2.1 ∘ ‖ ‖‖ ‖ sin Page 91 of 109 ‖ ‖ ‖ ‖‖ ‖ April 10, 2017 Chapter 1 Vectors Vector Triple Products Scalar Triple Product Let: u u , u v v , v w w w . The the triple product ∘ x gives a scalar represe ti g the volu e of a parallelepiped a D parallelogra ith , , a d as edges: ∘ ∘ u v w x x Note: vectors , , a d x u v w ∘ u v w are copla ar if a d o ly if ∘ x . Other Triple Products ∘ x x x ∘ x x x ∘ ∘ ∘ ∘ Duplicati g a vector results i a product of x x ∘ ∘ ∘ x No Associative Property The associative property of real u ∘ x Version 2.1 x ∙ ∙ x ∘ x ers does ot tra slate to triple products. I particular, No associative property of dot products/ ultiplicatio No associative property of cross products Page 92 of 109 April 10, 2017 Appe di A Su ary of Trigo o etric For ulas Appendix A Su ary of Trigo o etric For ulas Trigo o etric Fu ctio s ‐ a d ‐ axes sin θ sin θ cos θ cos θ tan θ tan θ cot θ cot θ sec θ sec θ csc θ Si e‐Cosi e Relatio ship Pythagorea Ide tities for a y a gle θ sin cos sec csc sin θ sin θ tan cot Cofu ctio s i Quadra t I cos ⇔ cos sec csc ⇔ csc Version 2.1 cot ⇔ cos θ cos θ sec θ cot θ tan θ tan θ cot θ cot sin sin θ cos θ cos θ sin θ cos θ sin θ Key A gles ° ° radians ° radians ° ° sin tan csc θ csc θ 9 ° radians radians radians tan sec Page 93 of 109 April 10, 2017 Appe di A Su ary of Trigo o etric For ulas Trigo o etric Fu ctio s Right Tria gle SOH‐CAH‐TOA sin sin tan tan cos cos sin cos tan Laws of Si es a d Cosi es O li ue Tria gle Law of Si es see illustratio elo Law of Cosi es see illustratio elo cos cos cos A c B Version 2.1 b C a Page 94 of 109 April 10, 2017 Appe di A Su ary of Trigo o etric For ulas A gle Additio For ulas sin sin sin cos sin cos tan cos sin cos sin cos cos cos cos cos cos sin sin sin sin tan Dou le A gle For ulas sin sin cos cos tan Half A gle For ulas cos cos sin The use of a + or ‐ sin sig i the half a gle for ulas depe ds o the uadra t i sin the a gle resides. See chart elo . Sig s of Trig Fu ctio s By Quadra t cos tan Version 2.1 hich Page 95 of 109 si + cos ‐ ta ‐ si + cos + ta + si ‐ cos ‐ ta + si ‐ cos + ta – April 10, 2017 Appe di A Su ary of Trigo o etric For ulas Power Reduci g For ulas sin cos tan Product‐to‐Su For ulas ∙ ∙ ∙ ∙ Su ‐to‐Product For ulas ∙ ∙ ∙ ∙ ∙ ∙ ∙ Version 2.1 ∙ Page 96 of 109 April 10, 2017 Appe di A Su ary of Trigo o etric For ulas Tria gle Area For ulas Geo etry here, is the le gth of the ase of the tria gle. is the height of the tria gle. Hero ’s For ula . here, , , are the le gths of the sides of the tria gle. Usi g Both Le gths a d A gles ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙ Coordi ate Geo etry Let three vertices of a tria gle i the coordi ate pla e e: , , , , , . ∙ Version 2.1 Page 97 of 109 April 10, 2017 Appe di A Su Co plex Nu cos Operatio s sin cis cos sin Let: cos Multiplicatio : ∙ Divisio : Po ers: Roots: √ cos varies fro has ers i Polar For ∙ sin sin cos cos Note: Version 2.1 cis ary of Trigo o etric For ulas cos sin sin sin , sin √ ∙ cos to disti ct co ple ‐th roots: Page 98 of 109 , , ,…, April 10, 2017 Appe di A Su ary of Trigo o etric For ulas Vectors , , are the u it vectors i the , , directio s respectively. di e sio s ‖ ‖ di e sio s ‖ ‖ √ √ Properties Additive Ide tity Additive I verse Co utative Property Associative Property Associative Property Distri utive Property Distri utive Property Multiplicative Ide tity ‖ ‖ ‖ ‖ Version 2.1 | |‖ ‖ Mag itude Property U it vector i the directio of Page 99 of 109 April 10, 2017 Appe di A Su ary of Trigo o etric For ulas Vector Dot Product Let: a d ∘ ∙ ∙ ∙ Properties ∘ )ero Property ∘ ∘ ∘ ∘ ‖ ‖ ∘ ∘ cos ∘ ∘ , a d ∘ Co are orthogo al to each other. utative Property Mag itude S uare Property ∘ ∘ ∘ ∘ Distri utive Property ∘ Multiplicatio y a Scalar Property is the a gle et ee ‖ ‖‖ ‖ a d Vector Projectio ∘ ∘ proj ‖ ‖ ∘ Orthogo al Co po e ts of a Vector proj Work ∘ ‖ ‖ a d F is the force vector acti g o a o ject, ∘ ‖ ‖ Version 2.1 cos ovi g it fro poi t is a gle et ee F a d Page 100 of 109 to poi t . . April 10, 2017 Appe di A Su ary of Trigo o etric For ulas Vector Cross Product 2 Di e sio s Let: u The , a d u u v x Area of a parallelogra u v v v u v u v a d v havi g a d as adjace t sides a d a gle θ et ee the : ‖ ‖ ‖ ‖ sin θ Di e sio s Let: u u v x u v u u u v u v ‖ ‖ ‖ ‖ sin x u v v v u v u v u v u v is the u it vector or al to the pla e co tai i g the first t o vectors ith orie tatio deter i ed usi g the right ha d rule. Properties x x , x x x sin , )ero Property x x x x ‖ x x x m ‖ ‖ ‖‖ ‖ Version 2.1 , x , , a d are orthogo al to each other Reverse orie tatio orthogo ality x Every o ‐zero vector is parallel to itself A ti‐co x x m x Distri utive Property x x m utative Property Distri utive Property x Scalar Multiplicatio is the a gle et ee Page 101 of 109 a d April 10, 2017 Appe di A Su ary of Trigo o etric For ulas Vector Triple Products Let: u u u , v v v , w w w . Scalar Triple Product ∘ ∘ u v w x x x u v w ∘ u v w Other Triple Products ∘ x x x ∘ x x x ∘ ∘ ∘ ∘ x x ∘ ∘ ∘ x No Associative Property ∘ x x ∙ Version 2.1 ∙ x ∘ x Page 102 of 109 April 10, 2017 Appe di B Solvi g the A iguous Case – Alter ative Method Appe dix B Solvi g the A iguous Case – Alter ative Method Ho do you solve a tria gle or t o i the a iguous case? Assu e the i for atio give is the le gths of sides a d , a d the easure of A gle . Use the follo i g steps: Step : Calculate the si e of the issi g a gle i this develop e t, a gle Step 2: Co sider the value of :  If , the  If , the  If , the Step : Co pare   e have Case 1 – there is o tria gle. Stop here. . Step : Use sin sin 9 °, a d e have Case – a right tria gle. Proceed to Step 4. e have Case or Case 4. Proceed to the e t step to deter i e hich. a d . , the e have Case – t o tria gles. Calculate the values of each a gle , usi g the If La of Si es. The , proceed to Step 4 a d calculate the re ai i g values for each tria gle. If Version 2.1 , the e have case 4 – o e tria gle. Proceed to Step 4. Page 103 of 109 April 10, 2017 Appe di B Solvi g the A iguous Case – Alter ative Method Step : Calculate . At this poi t, e have the le gths of sides a d , a d the easures of A gles a d . If e are deali g ith Case – t o tria gles, e ust perfor Steps 4 a d for each a gle. Step 4 is to calculate the easure of A gle as follo s: ° Step : Calculate . Fi ally, e calculate the value of usi g the La of Si es. sin sin Note: usi g A ⇒ a d∠ sin sin ay produce or sin sin Is ? ⇒ sin sin ore accurate results si ce oth of these values are give . iguous Case Alter ative Method Flowchart Start Here Value of sin yes no Two triangles Calculate , and then . Steps and , above Version 2.1 Page 104 of 109 April 10, 2017 Appe di C Su ary of Recta gular a d Polar For s Appendix C Su ary of Recta gular a d Polar For s Recta gular For Coordi ates For Co versio Co plex Nu ers For Co versio , Polar For , cos sin cos cos sin Vectors tan sin or tan For ‖ ‖ ‖ ‖∠ ag itude directio a gle Co versio ‖ ‖ cos ‖ ‖ sin Version 2.1 Page 105 of 109 ‖ ‖ tan April 10, 2017 Trigonometry Handbook Index Subject Page 4, 1 A iguous Case for O li ue Tria gles ,1 4 A iguous Case for O li ue Tria gles ‐ Flo chart 17 A plitude A gle 7 A gle Additio For ulas 14 A gle of Depressio 14 A gle of Depressio 7 Arc Measure 1 1 7 7 ,7 Area of a Tria gle Geo etry For ula Hero 's For ula Trigo o etric For ulas Coordi ate Geo etry For ula Astroid Cardioid 17 71 Characteristics of Trigo o etric Fu ctio Graphs Circles 9 ,1 Cofu ctio s Co ple Nu ers Co versio et ee Recta gular a d Polar For s Operatio s i Polar For Polar For 79 Co po e ts of Vectors ,1 4, 1 7 1 Co versio et ee Recta gular a d Polar For s Co ple Nu ers Coordi ates E uatio s Vectors 11 Coseca t Fu ctio 11 Cosi e Fu ctio 11 Cota ge t Fu ctio 9 Coter i al A gle Cross Product 7 Cycloid Version 2.1 Page 106 of 109 April 10, 2017 Trigonometry Handbook Index Page Subject 11 Defi itio s of Trig Fu ctio s Right Tria gle 9 Defi itio s of Trig Fu ctio s ‐ a d y‐ a es 7 Degrees 7 Deltoid 7 DeMoivre's Theore DeMoivre's Theore 14 for Roots Depressio , A gle of Dot Product 7 Dou le A gle For ulas 77 Ellipse 7 E uatio s Co versio 47 17 et ee Recta gular a d Polar For s Solvi g Trigo o etric E uatio s Fre ue cy 1 77 Graphs Basic Trig Fu ctio s Cardioid Coseca t Fu ctio Cosi e Fu ctio Cota ge t Fu ctio I verse Trigo o etric Fu ctio s Li aço of Pascal Polar Fu ctio s Rose Seca t Fu ctio Si e Fu ctio Ta ge t Fu ctio Trig Fu ctio Characteristics Ta le Half A gle For ulas Har o ic Motio Hero 's For ula Hyper ola 4 44 Ide tities ‐ Verificatio Steps Tech i ues 1 7 4 7 9 74 4 19 7 Version 2.1 Page 107 of 109 April 10, 2017 Trigonometry Handbook Index Page 14 Subject I cli atio , A gle of I itial Side of a A gle 4 I verse Trigo o etric Fu ctio s Defi itio s Graphs Pri cipal Values Ra ges La of Cosi es La of Si es 7 Le 7 Li aço of Pascal 7 Nephroid 1 iscate of Ber oulli O li ue Tria gle ‐ Methods to Solve Operatio s o Co ple Nu ers i Polar For Orthogo al Co po e ts of a Vector 77 Para ola 17 Period 1 Phase Shift Polar A gle , 9 4, 1 ,1 71 4, 1 41 Polar A is Polar Coordi ates Polar For of Co ple Nu Polar Graph Types ers Polar to Recta gular Coordi ate Co versio Po er Reduci g For ulas Pri cipal Values of I verse Trigo o etric Fu ctio s 41 Product‐to‐Su For ulas Projectio of O e Vector o to A other Properties of Vectors 9 Pythagorea Ide tities Quadra tal A gle 7, 9 4, Radia s Recta gular to Polar Coordi ate Co versio Version 2.1 Page 108 of 109 April 10, 2017 Trigonometry Handbook Index Page Subject Refere ce A gle 71, 74 11 Rose Seca t Fu ctio Si ple Har o ic Motio 11 9 Si e Fu ctio Si e‐Cosi e Relatio ship 11 SOH‐CAH‐TOA 7 Spirals Sta dard Positio 41 Su ‐to‐Product For ulas 11 Ta ge t Fu ctio Ter i al Side of a A gle 1 Trigo o etric Fu ctio Values i Quadra ts II, III, a d IV 11 Trigo o etric Fu ctio s of Special A gles 9 Triple Products 1 U it Circle 79 U it Vectors ‐ i a d j 79 79 1 9 Vectors Co po e ts Co versio et ee Recta gular a d Polar For s Cross Product Dot Product Orthogo al Co po e ts of a Vector Projectio Properties Special U it Vectors ‐ i a d j Triple Products 9 Verte of a A gle 1 Vertical Shift Work Version 2.1 Page 109 of 109 April 10, 2017