Jay Abramson

Review Exercises

Simplifying and Verifying Trigonometric 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.

$\sin (70^{\circ}) \cos (25^{\circ}) - \cos (70^{\circ}) \sin (25^{\circ})$

14.

$\cos (83^{\circ}) \cos (23^{\circ}) + \sin (83^{\circ}) \sin (23^{\circ})$

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 zeroes on the interval $[0, 2 π) .$

48.

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

49.

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

Modeling with Trigonometric Equations

For the following exercises, graph the points and find a possible formula for the trigonometric values in the given table.

50.

$x$	$y$
$0$	$1$
$1$	$6$
$2$	$11$
$3$	$- 6$
$4$	$- 1$
$5$	$6$

51.

$x$	$y$
$0$	$- 2$
$1$	$1$
$2$	$- 2$
$3$	$- 5$
$4$	$- 2$
$5$	$1$

52.

$x$	$y$
$- 3$	$3 + 2 \sqrt{2}$
$- 2$	$3$
$- 1$	$2 \sqrt{2} - 1$
$0$	$1$
$1$	$3 - 2 \sqrt{2}$
$2$	$- 1$
$3$	$−1 - 2 \sqrt{2}$

53.

A man with his eye level 6 feet above the ground is standing 3 feet away from the base of a 15-foot vertical ladder. If he looks to the top of the ladder, at what angle above horizontal is he looking?

54.

Using the ladder from the previous exercise, if a 6-foot-tall construction worker standing at the top of the ladder looks down at the feet of the man standing at the bottom, what angle from the horizontal is he looking?

For the following exercises, construct functions that model the described behavior.

55.

A population of lemmings varies with a yearly low of 500 in March. If the average yearly population of lemmings is 950, write a function that models the population with respect to $t,$ the month.

56.

Daily temperatures in the desert can be very extreme. If the temperature varies from $90 °F$ to $30 °F$ and the average daily temperature first occurs at 10 AM, write a function modeling this behavior.

For the following exercises, find the amplitude, frequency, and period of the given equations.

57.

$y = 3 \cos (x π)$

58.

$y = −2 \sin (16 x π)$

For the following exercises, model the described behavior and find requested values.

59.

An invasive species of carp is introduced to Lake Freshwater. Initially there are 100 carp in the lake and the population varies by 20 fish seasonally. If by year 5, there are 625 carp, find a function modeling the population of carp with respect to $t,$ the number of years from now.

60.

The native fish population of Lake Freshwater averages 2500 fish, varying by 100 fish seasonally. Due to competition for resources from the invasive carp, the native fish population is expected to decrease by 5% each year. Find a function modeling the population of native fish with respect to $t,$ the number of years from now. Also determine how many years it will take for the carp to overtake the native fish population.