Trigonometric Identities

Transcription

1 Trigonometric Identities 6.5 SKILL BUILDER An equation that is true for all alues of the ariable in it is called an identit. For instance, the epression 4( 3) 8 is an eample of an algebraic identit because it is true for all alues of. Both sides of the epression are equialent, or identical. If each side of the equation were separated and graphed, both graphs would be identical = 4( + 3) = Some trigonometric equations can also be identities. Howeer, it is not alwas obious that both sides represent identical epressions. Showing that both sides of the equation represent the same epression proes that the original equation is an identit. Fundamental Trigonometric Identities Recall that for an angle,, in standard pition, where P (, ) is a point on the terminal arm of the angle, the primar trigonometric rati are r c r tan P(, ) r 0 The Quotient Identit Eamining the ratio of allows ou to deelop an equialent epression. c c r r r r Howeer, tan. Therefore, c tan ➀ Equation ➀ is called a quotient identit. Both sides of this identit produce eactl the same graph. 6.5 TRIGONOMETRIC IDENTITIES 535

2 3 3 -π -π - π π π -π - π π - -3 c si n tan For an angle, the two epressions produce the eact same alues, with the eception of where the functions are undefined. Therefore, the equation has an infinite number of solutions. A Pthagorean Identit For an angle in standard pition, r. P(, ) Because r is the distance from the origin to P, r 0. Diide the equation b r to deelop an equialent epression. r r r r r r r r r r ( ) (c ) c ➁ Equation ➁ is called a Pthagorean identit. Both sides of this identit produce eactl the same graph. -π - 3π -π - π 0 π π 3π π - -π - 3π -π - π 0 π π 3π π - c For an angle,, the two epressions produce eactl the same alue. Therefore, the equation has an infinite number of solutions. Rearranging this identit gies two other ersions. c and c The quotient identit and this Pthagorean identit are often called the fundamental trigonometric identities because the can be used to proe that other more complicated equations are also identities. To proe that an equation is an identit, ou must simplif the epression and show that the left side equals the right side. 536 CHAPTER 6 EXTENDING SKILLS WITH TRIGONOMETRY

3 It is often best to simplif the more complicated side first and rewrite epressions in terms of e and coe. Eample Proe tan c. Eample Proe that tan c. tan c c c, therefore for all, tan c. tan (tan ) c c ( ) (c ) ( ) c ( ) ( c ) c, therefore for all, tan c. Often, it is necessar to find a common denominator and then add or subtract epressions. In this case, alwas look for epressions inoling or c. These are cases where the Pthagorean identit, c, can be used. Sometimes it is necessar to factor, as Eample 4 shows. Eample 3 Proe that tan tan. c tan ta n c c c c c c c c c c c c Eample 4 Proe that c. c c c c ( c )( c ) c c, therefore, c c for all. c, therefore tan tan for all alues of. c 6.5 TRIGONOMETRIC IDENTITIES 537

4 Ke Ideas An identit is an mathematical equation that is true for all alues of the ariable. For eample, 3( ) 3 3 is an identit. A trigonometric identit is an mathematical equation with trigonometric epressions that is true for all alues of the ariable. Fundamental Trigonometric Identities Quotient Identit: c tan Pthagorean Identit: c c c It is not alwas obious that both sides of a trigonometric epression are equal. To proe that it is an identit, a proof that shows that both sides of the epression are equal is required. To proe that a gien epression is an identit follow these steps.. Separate the two sides of the epression.. Simplif the more complicated side until it is identical to the other side or transform both sides of the epression into the same epression. The following strategies can be helpful for proing identities.. Epress all tangent functions in terms of the e function or the coe function.. Look for epressions to which the Pthagorean identit can be applied. 3. Where necessar, factor or find a common denominator. It is not alwas eas to proe an identit. If ou get stuck or take a wrong turn, tr another approach. Practise, Appl, Sole 6.5 A. Use the definitions, c r, and tan to proe each identit. (a) c tan r (b) tan co s (c) c 538 CHAPTER 6 EXTENDING SKILLS WITH TRIGONOMETRY

5 . Simplif. (a) c (b) (c )(tan ) (c) c (d) (e) c (f) ( )( ) (g) t an (h) (i) tan (j) tan tan (m) co s c tan c (k) s in (n) tan co (l) c c (o) s tan 3. Factor each epression. (a) c (b) (c) c (d) (e) c c (f) c a 4. (a) Proe that t an a a, b epresg the left side in terms of a a. (b) Proe the identit. c 5. (a) Proe that b epresg c in terms of. (b) Factor the epression in (a). (c) Proe the identit. B 6. Verif each identit. (a) t c an (b) t an c (c) co a s tan a a c a (d) c tan c 7. Proe each identit. (a) c tan a (b) tan c a a (c) c ( )( ) (d) c c 6.5 TRIGONOMETRIC IDENTITIES 539

6 8. Proe each identit. (a) ( c ) c (b) co s 4 (c) c tan co (d) c tan 9. Proe each identit. (a) tan s tan (b) tan tan (c) tan c co c s (d) c c 0. Which equations are not identities? Justif our conclusion. (a) (b) c c c b c b c b b tan tan b c b (c) ( c )( tan ) (d) c 4 c 4. Proe each identit. (a) c tan 3 tan (b) c 4 c 4 (c) ( c ) tan tan co (d) tan b c b b c b c c tan. Proe that. s 3. Check Your Understanding: Proe that tan. C 4. Proe each identit. tan tan tan c c (a) c b b b c b (b) c tan f c f f 3 c f 3 4 c f c f f 5. Proe that. f 4 f f 6. Proe that f c f tan f. 540 CHAPTER 6 EXTENDING SKILLS WITH TRIGONOMETRY