Jay Abramson

Review Exercises

Solving Trigonometric Equations with Identities

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

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.

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.

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

For the following exercises, find the exact value.

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.

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.

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.

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

For the following exercises, find the exact value.

20.

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

21.

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

22.

$\sin (\frac{7 π}{8})$

23.

$\sec (\frac{3 π}{8})$

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

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. — Figure 1

24.

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

25.

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

For the following exercises, prove the identity.

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.

28.

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

29.

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

Sum-to-Product and Product-to-Sum Formulas

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

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.

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.

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

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

39.

$\tan x + 1 = 0$

40.

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

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

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.

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 $[0, 2 π) .$

48.

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

49.

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