7 Chapter Test

Simplify.

1. $\frac{2 {cos}^{2} x - cos x - 1}{cos x - 1}$ $\frac{2 {cos}^{2} x - cos x - 1}{cos x - 1}$
2. ${(\frac{sec x}{tan x})}^{2} - \frac{1}{{tan}^{2} x}$ ${(\frac{sec x}{tan x})}^{2} - \frac{1}{{tan}^{2} x}$
3. Rationalize the denominator:

$\sqrt{\frac{1 - sin θ}{1 + sin θ} .}$ $\sqrt{\frac{1 - sin θ}{1 + sin θ} .}$

Assume that the radicand is nonnegative.
4. Given that $x = 2 sin θ$ $x = 2 sin θ$ , express $\sqrt{4 - x^{2}}$ $\sqrt{4 - x^{2}}$ as a trigonometric function without radicals. Assume $0 < θ < π / 2$ $0 < θ < π / 2$ .

Use the sum or difference identities to evaluate exactly.

5. $sin 75 °$ $sin 75 °$
6. $tan \frac{π}{12}$ $tan \frac{π}{12}$
7. Assuming that $cos u = \frac{5}{13}$ $cos u = \frac{5}{13}$ and $cos v = \frac{12}{13}$ $cos v = \frac{12}{13}$ and that $u$ $u$ and $v$ $v$ are between 0 and $π / 2$ $π / 2$ , evaluate $cos (u - v)$ $cos (u - v)$ exactly.
8. Given that $cos θ = - \frac{2}{3}$ $cos θ = - \frac{2}{3}$ and that the terminal side is in quadrant II, find $cos (π / 2 - θ)$ $cos (π / 2 - θ)$ .
9. Given that $sin θ = - \frac{4}{5}$ $sin θ = - \frac{4}{5}$ and $θ$ $θ$ is in quadrant III, find $sin 2 θ$ $sin 2 θ$ and the quadrant in which $2 θ$ $2 θ$ lies.
10. Use a half-angle identity to evaluate $cos \frac{π}{12}$ $cos \frac{π}{12}$ exactly.
11. Given that $sin θ = 0.6820$ $sin θ = 0.6820$ and that $θ$ $θ$ is in quadrant I, find $cos (θ / 2)$ $cos (θ / 2)$ .
12. Simplify: ${(sin x + cos x)}^{2} - 1 + 2 sin 2 x$ ${(sin x + cos x)}^{2} - 1 + 2 sin 2 x$ .

Prove each of the following identities.

13. $csc x - cos x cot x = sin x$ $csc x - cos x cot x = sin x$
14. ${(sin x + cos x)}^{2} = 1 + sin 2 x$ ${(sin x + cos x)}^{2} = 1 + sin 2 x$
15. ${(csc β + \cot β)}^{2} = \frac{1 + cos β}{1 - cos β}$ ${(csc β + \cot β)}^{2} = \frac{1 + cos β}{1 - cos β}$
16. $\frac{1 + sin α}{1 + csc α} = \frac{tan α}{sec α}$ $\frac{1 + sin α}{1 + csc α} = \frac{tan α}{sec α}$

Use the product-to-sum identities and the sum-to-product identities to find identities for each of the following.

17. $cos 8 α - cos α$ $cos 8 α - cos α$
18. $4 sin β cos 3 β$ $4 sin β cos 3 β$
19. Find ${sin}^{- 1} (- \frac{\sqrt{2}}{2})$ ${sin}^{- 1} (- \frac{\sqrt{2}}{2})$ exactly in degrees.
20. Find ${tan}^{- 1} \sqrt{3}$ ${tan}^{- 1} \sqrt{3}$ exactly in radians.
21. Use a calculator to find ${cos}^{- 1} (- 0.6716)$ ${cos}^{- 1} (- 0.6716)$ in radians, rounded to four decimal places.
22. Evaluate $cos ({sin}^{- 1} \frac{1}{2})$ $cos ({sin}^{- 1} \frac{1}{2})$ .
23. Find $tan ({sin}^{- 1} \frac{5}{x})$ $tan ({sin}^{- 1} \frac{5}{x})$ .
24. Evaluate $cos ({sin}^{- 1} \frac{1}{2} + {cos}^{- 1} \frac{1}{2})$ $cos ({sin}^{- 1} \frac{1}{2} + {cos}^{- 1} \frac{1}{2})$ .

Solve, finding all solutions in $[0, 2 π)$ $[0, 2 π)$ .

25. $4 {cos}^{2} x = 3$ $4 {cos}^{2} x = 3$
26. $2 {sin}^{2} x = \sqrt{2} sin x$ $2 {sin}^{2} x = \sqrt{2} sin x$
27. $\sqrt{3} cos x + sin x = 1$ $\sqrt{3} cos x + sin x = 1$
28. The graph of $f (x) = {cos}^{- 1} x$ $f (x) = {cos}^{- 1} x$ is which of the following?

Synthesis

29. Find $cos θ$ $cos θ$ , given that $cos 2 θ = \frac{5}{6}, \frac{3 π}{2} < θ < 2 π$ $cos 2 θ = \frac{5}{6}, \frac{3 π}{2} < θ < 2 π$ .

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