Verifying Trigonometric Identities

Transcription

1 40 Chapter Analytic Trigonometry. f x sec x Sketch the graph of y cos x Amplitude: Period: One cycle: first. The x-intercepts of y correspond to the vertical asymptotes of f x. cos x sec x 4 x, x 4 4,... x 0 x 4 4 x 4 x π 4 y π π x 4 4. f x cosx Using y a cos bx, a so the amplitude is b so the period is. x shifts the graph right by and shifts the graph upward by.. y 4 π π π x Section. Verifying Trigonometric Identities You should know the difference between an expression, a conditional equation, and an identity. You should be able to solve trigonometric identities, using the following techniques. (a) Work with one side at a time. Do not cross the equal sign. Use algebraic techniques such as combining fractions, factoring expressions, rationalizing denominators, and squaring binomials. Use the fundamental identities. (d) Convert all the terms into sines and cosines. Vocabulary Check. identity. conditional equation. tan u 4. cot u. cos u 6. sin u 7. csc u 8. sec u. csc t. sec y cos y cos y cos y. sin sin sin cos 4. cot ysec y cot y tan y

2 Section. Verifying Trigonometric Identities 4. cos sin sin sin 6. cos sin cos cos sin cos 7. sin sin 4 sin sin 8. cos cos cos cos 4 tan x sin x sec x 9. csc csc cot cot 0. csc tan sin sin cos sin cos csc sec cot t csc t cot t cot t csc t cot tcsc t csc t cos t csc t cos t csc t cos tcsc t. cot t csc t cos t. sin t cos t csc t tan tan tan tan sec tan. sin x sin x sin x sin x sin x cos x cos x 4. sec 6 xsec x tan x sec 4 xsec x tan x sec 4 xsec x tan xsec x sec 4 xsec x tan x tan x sec x tan x. cot x sec x tan x csc x 6. sec sec cos sec sec sec sec sec sec sec

3 4 Chapter Analytic Trigonometry 7. csc x 8. sec x cot x tan x 9. tan x cot x tan x cot x tan x cot x cot x tan x tan x cot x 0. csc x csc x csc x csc x csc x. cos cot cos cot sin sin sin. sin cos sin cos cos sin cos sin cos cos sin csc sin sin sin cos sin sin sin sin sin sin sin sin sin sin sin cos cos sin sin cos sin sin cos sin cos sec. csc x csc x csc x csc x csc x csc x csc x csc x csc x csc x 4. tan x tan x tan x tan x tan x. tan tan cot tan tan tan 6. cos x tan x 7. sin x cscx sinx secx cosx cosx sinx cot x

4 Section. Verifying Trigonometric Identities 4 8. sin y siny sin y sin y 9. sin y cos y tan x cot x sec x 0. tan x tan y tan x tan y cot x cot y cot x cot y cot y cot x cot x cot y cot x cot y cot x cot y. tan x cot y tan x cot y cot x tan y cot x tan y tan y cot x cot x tan y cot x tan y. cos y sin y cos y cos y sin y sin y sin y cos y sin y cos y cos y sin x sin y sin y cos y sin x cos y sin y sin y cos y 0. sin sin sin sin sin sin 4. cos cos cos cos cos cos sin sin cos cos sin cos cos sin sin cos cos sin. cos cos cos sin 6. sec y cot y sec y tan y 7. csc t sec t cos t tan t cos t 8. sec x csc x cot x 9. (a) Let y and y. CONTINUED

5 44 Chapter Analytic Trigonometry 9. CONTINUED sec x sec x sin x sin x cos x sec x sin x sin x cos x sec xcos x cos x 40. (a) csc xcsc x csc x cot x cot x csc x cot x csc x csc x cot x 4. (a) y y Let y 4 and y. cos x cos 4 x cos x cos x sin x cos x sin x cos x 4. (a) tan 4 x tan x sin4 x cos 4 x cos x cos x sin4 x cos x cos x sin4 x sin x cos x cos x cos x cos x cos x cos x cos x sec x tan x sec x4 tan x

6 Section. Verifying Trigonometric Identities 4 4. (a) 44. (a) Let y and 4 y tan x 4. csc 4 x csc x csc x cot x cot 4 x sin 4 sin cos sin cos cos cos cos 4. (a) 46. (a) y y Let y and y. sin x cos x cot csc is the reciprocal of csc. cot They will only be equivalent at isolated points in their respective domains. Hence, not an identity. 47. tan x sec x tan x tan xsec x 48. tan x tan x tan x tan x tan 4 x sec x sin x cos x sin4 x cos 4 x cos x cos 4 x sin4 x cos x cos x sin 4 x 4 cos x cos x sin x 4 cos x cos 4 x cos x sec4 x tan x

7 46 Chapter Analytic Trigonometry 49. sin x sin 4 x sin x sin x 0. sin x cos x sin x cos x sin 4 x cos 4 x sin x sin x cos 4 x cos x cos x cos 4 x cos x cos 4 x cos 4 x cos x cos 4 x. sin sin 6 sin cos sin cos cos cos cos sin 90 cos sin. cos 0 cos cos 8 cos 70 cos 0 cos sin 90 8 sin cos 0 cos sin sin 0 cos 0 sin 0 cos sin 4. sin sin 40 sin 0 sin 78 sin sin 78 sin 40 sin 0 cos 90 sin 78 cos sin 0 cos 78 sin 78 cos 0 sin 0. csc x cot x sin x csc x csc x cot x 6. (a) h sin90 sin 0 h cos h cot sin Greatest: 0, Least: (d) Noon 90 s False. For the equation to be an identity, it must be true for all values of in the domain. 8. True. An identity is an equation that is true for all real values in the domain of the variable. 9. Since sin cos,60. then sin ± cos ; sin cos if such angle is 7 4. lies in Quadrant III or IV. One tan sec True identity: tan ±sec tan sec is not true for < < < <. Thus, the equation is not true for 4. or

8 Section. Verifying Trigonometric Identities i 6 i 6i 6. 6 i i i i 4 0i i 4 0i 0i i i 64. 4i 8i 4i 8 8 4i i i i i 9 i 4i i i i 0i 6i 4i 9 46i 6. x 6x a, b 6, c x 6 ± ± ± 84 6 ± ± x x 7 0 a, b, c 7 x ± 47 x ± 67. x 6x x x 4 0 x x 4 0 a, b, c 4 x ± 44 ± 4 6 ± 0 ± ± 8x 4x 0 a 8, b 4, c x 4 ± ± 6 x 4 ± 47 6 x ± 7 4