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1 646 Chapter 7 Analytic Trigonometry. f x x Amplitude: Period: Key points: 0, 0,,,, 0,,,, 0 y x. f x tan x Amplitude: Period: Two consecutive vertical asymptotes: x, x Key points:,, 0, 0,, y x. f x sec x Sketch the graph of y x Amplitude: Period: One cycle: first. The x-intercepts of y correspond to the vertical asymptotes of f x. 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 x y Ug y a bx, a so the amplitude is. 4 b so the period is. π π π x x shifts the graph right by and shifts the graph upward by. Section 7. 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, ug 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 es and ines.
2 Section 7. Verifying Trigonometric Identities 647 Vocabulary Check. identity. conditional equation. tan u 4. cot u. u 6. u 7. csc u 8. sec u. t csc t t t. sec y y y y. 4. cot ysec y cot y tan y tan x x sec x 9. csc cot csc csc csc cot sec tan 0. cot t csc t cot t cot t csc t cot tcsc t csc t t t csc t t t t csc t t tcsc t. cot t csc t t t. t t t t t csc t t t t t tan tan tan tan sec tan. x x x x x x x
3 648 Chapter 7 Analytic Trigonometry 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 sec sec sec sec sec sec sec 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. cot csc cot. sec
4 Section 7. Verifying Trigonometric Identities 649. 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. x tan x x 7. cscx x secx x x x cot x 8. y y y y y y 9. 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. y y y y y y y y y y y x y y y x y y y y 0. 4.
5 60 Chapter 7 Analytic Trigonometry. 6. sec y cot y sec y tan y 7. t csc t t sec t t t t tan t t 8. sec x csc x cot x 9. (a) Let y and y. sec x sec x x x x sec x x x x sec x x x 40. (a) csc xcsc x cot x csc x csc x cot x csc x cot x cot x csc x 4. (a) y y Let y 4 and y. x 4 x x x x x x x
6 Section 7. Verifying Trigonometric Identities 6 4. (a) tan 4 x tan x 4 x 4 x x x 4 x x x 4 x x x x x x x x x sec x tan x sec x4 tan x 4. (a) Let y and 4 y tan x 4. csc 4 x csc x csc x cot x cot 4 x 44. (a) 4
7 6 Chapter 7 Analytic Trigonometry 4. (a) y y Let y and y x x. 46. (a) 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 x x 4 x 4 x x 4 x 4 x x x 4 x 4 x x x 4 x 4 x x sec4 x tan x 49. x 4 x x x 0. x x x x 4 x 4 x x x 4 x x x 4 x x 4 x 4 x x 4 x
8 Section 7. Verifying Trigonometric Identities csc x cot x x csc x csc x cot x 6. (a) h 90 0 h 0 h cot 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,60. then ± ; Quadrant III or IV. One such angle is 7 4. if lies in tan sec True identity: tan sec tan is not true for < < < <. Thus, the equation is not true for 4. ±sec or 6. i 6 i 6i 6i 6. 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
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