第9章三角関数の恒等式と方程式Trigonometric Identities and Equations

Review Exercises復習問題

Simplifying and Verifying Trigonometric Identities三角関数の恒等式を簡単にして確かめる

For the following exercises, find all solutions exactly that exist on the interval $[0, 2 π) .$

次の各問について、区間 $[0, 2 π)$ にある解をすべて厳密に求めよ。

1. $\csc^{2} t = 3$

2. $\cos^{2} x = \frac{1}{4}$

3. $2 \sin θ = −1$

4. $\tan x \sin x + \sin (- x) = 0$

5. $9 \sin ω - 2 = 4 \sin^{2} ω$

6. $1 - 2 \tan (ω) = \tan^{2} (ω)$

For the following exercises, use basic identities to simplify the expression.

次の各問について、基本恒等式を使って式を簡単にせよ。

7. $\sec x \cos x + \cos x - \frac{1}{\sec x}$

8. $\sin^{3} x + \cos^{2} x \sin x$

For the following exercises, determine if the given identities are equivalent.

次の各問について、与えられた等式が同値かどうかを判定せよ。

9. $\sin^{2} x + \sec^{2} x - 1 = \frac{(1 - \cos^{2} x) (1 + \cos^{2} x)}{\cos^{2} x}$

10. $\tan^{3} x \csc^{2} x \cot^{2} x \cos x \sin x = 1$

Sum and Difference Identities加法定理

For the following exercises, find the exact value.

次の各問について、厳密な値を求めよ。

11. $\tan (\frac{7 π}{12})$

12. $\cos (\frac{25 π}{12})$

13. $\sin (70°) \cos (25°) - \cos (70°) \sin (25°)$

14. $\cos (83°) \cos (23°) + \sin (83°) \sin (23°)$

For the following exercises, prove the identity.

次の各問について、等式を証明せよ。

15. $\cos (4 x) - \cos (3 x) \cos x = \sin^{2} x - 4 \cos^{2} x \sin^{2} x$

16. $\cos (3 x) - \cos^{3} x = - \cos x \sin^{2} x - \sin x \sin (2 x)$

For the following exercise, simplify the expression.

次の問について、式を簡単にせよ。

17. $\frac{\tan (\frac{1}{2} x) + \tan (\frac{1}{8} x)}{1 - \tan (\frac{1}{8} x) \tan (\frac{1}{2} x)}$

For the following exercises, find the exact value.

次の各問について、厳密な値を求めよ。

18. $\cos (\sin^{- 1} (0) - \cos^{- 1} (\frac{1}{2}))$

19. $\tan (\sin^{- 1} (0) + \sin^{- 1} (\frac{1}{2}))$

Double-Angle, Half-Angle, and Reduction Formulas倍角・半角・次数を下げる公式

For the following exercises, find the exact value.

次の各問について、厳密な値を求めよ。

20. Find $\sin (2 θ), \cos (2 θ),$ and $\tan (2 θ)$ given $\cos θ = - \frac{1}{3}$ and $θ$ is in the interval $[\frac{π}{2}, π] .$

20. $\cos θ = - \frac{1}{3}$ で $θ$ が区間 $[\frac{π}{2}, π]$ にあるとき、 $\sin (2 θ), \cos (2 θ)$ と $\tan (2 θ)$ を求めよ。

21. Find $\sin (2 θ), \cos (2 θ),$ and $\tan (2 θ)$ given $\sec θ = - \frac{5}{3}$ and $θ$ is in the interval $[\frac{π}{2}, π] .$

21. $\sec θ = - \frac{5}{3}$ で $θ$ が区間 $[\frac{π}{2}, π]$ にあるとき、 $\sin (2 θ), \cos (2 θ)$ と $\tan (2 θ)$ を求めよ。

22. $\sin (\frac{7 π}{8})$

23. $\sec (\frac{3 π}{8})$

For the following exercises, use Figure 1 to find the desired quantities.

次の各問について、図1を使って求められた量を求めよ。

Image of a right triangle. The base is 24, the height is unknown, and the hypotenuse is 25. The angle opposite the base is labeled alpha, and the remaining acute angle is labeled beta.

24. $\sin (2 β), \cos (2 β), \tan (2 β), \sin (2 α), \cos (2 α), and \tan (2 α)$

式中の英語：and：および

25. $\sin (\frac{β}{2}), \cos (\frac{β}{2}), \tan (\frac{β}{2}), \sin (\frac{α}{2}), \cos (\frac{α}{2}), and \tan (\frac{α}{2})$

式中の英語：and：および

For the following exercises, prove the identity.

次の各問について、等式を証明せよ。

26. $\frac{2 \cos (2 x)}{\sin (2 x)} = \cot x - \tan x$

27. $\cot x \cos (2 x) = - \sin (2 x) + \cot x$

For the following exercises, rewrite the expression with no powers.

次の各問について、累乗を含まない形に式を書き直せ。

28. $\cos^{2} x \sin^{4} (2 x)$

29. $\tan^{2} x \sin^{3} x$

Sum-to-Product and Product-to-Sum Formulas和積・積和の公式

For the following exercises, evaluate the product for the given expression using a sum or difference of two functions. Write the exact answer.

次の各問について、二つの関数の和または差を使って、与えられた式の積の値を求めよ。厳密な答えを書け。

30. $\cos (\frac{π}{3}) \sin (\frac{π}{4})$

31. $2 \sin (\frac{2 π}{3}) \sin (\frac{5 π}{6})$

32. $2 \cos (\frac{π}{5}) \cos (\frac{π}{3})$

For the following exercises, evaluate the sum by using a product formula. Write the exact answer.

次の各問について、積の公式を使って和の値を求めよ。厳密な答えを書け。

33. $\sin (\frac{π}{12}) - \sin (\frac{7 π}{12})$

34. $\cos (\frac{5 π}{12}) + \cos (\frac{7 π}{12})$

For the following exercises, change the functions from a product to a sum or a sum to a product.

次の各問について、関数を積から和に、または和から積に変えよ。

35. $\sin (9 x) \cos (3 x)$

36. $\cos (7 x) \cos (12 x)$

37. $\sin (11 x) + \sin (2 x)$

38. $\cos (6 x) + \cos (5 x)$

Solving Trigonometric Equations三角方程式を解く

For the following exercises, find all exact solutions on the interval $[0, 2 π) .$

次の各問について、区間 $[0, 2 π)$ の厳密な解をすべて求めよ。

39. $\tan x + 1 = 0$

40. $2 \sin (2 x) + \sqrt{2} = 0$

For the following exercises, find all exact solutions on the interval $[0, 2 π) .$

次の各問について、区間 $[0, 2 π)$ の厳密な解をすべて求めよ。

41. $2 \sin^{2} x - \sin x = 0$

42. $\cos^{2} x - \cos x - 1 = 0$

43. $2 \sin^{2} x + 5 \sin x + 3 = 0$

44. $\cos x - 5 \sin (2 x) = 0$

45. $\frac{1}{\sec^{2} x} + 2 + \sin^{2} x + 4 \cos^{2} x = 0$

For the following exercises, simplify the equation algebraically as much as possible. Then use a calculator to find the solutions on the interval $[0, 2 π) .$ Round to four decimal places.

次の各問について、方程式をできるかぎり代数的に簡単にせよ。そして電卓を使って区間 $[0, 2 π)$ の解を求めよ。小数第四位まで丸めよ。

46. $\sqrt{3} \cot^{2} x + \cot x = 1$

47. $\csc^{2} x - 3 \csc x - 4 = 0$

For the following exercises, graph each side of the equation to find the approximate solutions on the interval $[0, 2 π) .$

次の各問について、方程式の両辺のグラフを描いて、区間 $[0, 2 π)$ の近似の解を求めよ。

48. $20 \cos^{2} x + 21 \cos x + 1 = 0$

49. $\sec^{2} x - 2 \sec x = 15$