Q: Which of the following is the general solution for the equation $$\cos \theta = \frac{1}{2}$$?

$$\theta = \frac{\pi}{3} + 2k\pi or$$ $$\theta = \frac{5\pi}{3} + 2k\pi$$, $$k$$ any integer

Q: Solve the equation $$2\sin \theta - 1 = 0$$ for $$\theta in$$ $$[0, 2\pi).$$

$$\theta = \frac{\pi}{6}, \frac{5\pi}{6}$$

Q: Which of the following best describes the solution set for $$\sin 2\theta = 0 in$$ $$[0, 2\pi)$$?

$$\theta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$$

Q: Which of the following equations has no real solution?

$$\sin^2$ \theta + \cos^2$ \theta = 2$$

Q: Solve $$\tan(\theta) = 1$$ for $$\theta in$$ $$[0, 2\pi).$$

$$\theta = \frac{\pi}{4}, \frac{5\pi}{4}$$

Q: Which of the following is the correct general solution for $$\sin \theta = 0$$?

$$\theta = k\pi$$, $$k$$ any integer

Q: Which of the following is a solution to $$2\sin^2$ \theta - 1 = 0 in$$ $$[0, 2\pi)$$?

$$\theta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$$

Question 1

Which of the following is the general solution for the equation $$\cos 	heta = \frac{1}{2}$$?

Accepted Answer

$$	heta = \frac{\pi}{3} + 2k\pi or$$ $$	heta = \frac{5\pi}{3} + 2k\pi$$, $$k$$ any integer

Question 2

If $$\sin 	heta = -\frac{\sqrt{2}}{2}$$, which of the following is a solution for $$	heta in $$the interval $$[0, 2\pi)$$?

Accepted Answer

$$	heta = \frac{7\pi}{4}$$

Question 3

Solve the equation $$2\sin 	heta - 1 = 0$$ for $$	heta in$$ $$[0, 2\pi).$$

Accepted Answer

$$	heta = \frac{\pi}{6}, \frac{5\pi}{6}$$

Question 4

Which identity can be used to rewrite $$\sin^2$ 	heta + \cos^2$ 	heta$$?

Accepted Answer

$$1$$

Question 5

A medical device rotates with an angle $$	heta$$ such that $$	an 	heta = -2. $$Using a calculator in radians, which of the following is a solution for $$	heta in$$ $$[0, 2\pi) ($$rounded to two decimal places)?

Accepted Answer

$$	heta = 2\pi - 1.11$$

Question 6

Solve the quadratic trigonometric equation $$2\cos^2$ 	heta - \cos 	heta - 1 = 0$$ for $$	heta in$$ $$[0, 2\pi).$$

Accepted Answer

Question 7

Which of the following best describes the solution set for $$\sin 2	heta = 0 in$$ $$[0, 2\pi)$$?

Accepted Answer

$$	heta = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$$

Question 8

A chemical reaction's rate is modeled by $$y = 5\sin x + x. If$$ $$y = 3$$, which of the following methods is most appropriate to find $$x$$?

Accepted Answer

Use a graphing utility to find the intersection points of $$y_1 = 5$$\sin x + x$$ and y_2$ = 3$$

Question 9

Which of the following equations has no real solution?

Accepted Answer

$$\sin^2$ 	heta + \cos^2$ 	heta = 2$$

Question 10

Solve $$	an(	heta) = 1$$ for $$	heta in$$ $$[0, 2\pi).$$

Accepted Answer

$$	heta = \frac{\pi}{4}, \frac{5\pi}{4}$$

Question 11

Which of the following is a solution to $$\sin 	heta = 0.5 in$$ $$[0, 2\pi)$$?

Accepted Answer

$$	heta = \frac{\pi}{6}$$

Question 12

If $$\sin 	heta = 0$$ and $$	heta is in$$ $$[0, 2\pi)$$, what are the possible values for $$	heta$$?

Accepted Answer

$$	heta = 0, \pi$$

Question 13

A rotating platform completes one revolution every $$2\pi$$ seconds. If $$\cos 	heta = -1$$, at what time $$t (in $$seconds) does this occur for $$t in$$ $$[0, 2\pi)$$?

Accepted Answer

$$t = \pi$$

Question 14

Which of the following is the correct general solution for $$\sin 	heta = 0$$?

Accepted Answer

$$	heta = k\pi$$, $$k$$ any integer

Question 15

Solve $$\cos 2	heta = 1$$ for $$	heta in$$ $$[0, 2\pi).$$

Accepted Answer

$$	heta = 0, \pi$$

Question 16

Which of the following is a solution to $$2\sin^2$ 	heta - 1 = 0 in$$ $$[0, 2\pi)$$?

Accepted Answer

$$	heta = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}$$

Analytic Trigonometry: Solving Trigonometric Equations

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