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

$$\theta = \frac{\pi}{3} + 2k\pi,\ \theta = \frac{5\pi}{3} + 2k\pi,\ k \in \mathbb{Z}$$

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

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

Q: Which identity can be used to rewrite $$1 - 2\sin^2$ \theta in $$terms of $$\cos 2\theta$$?

$$\cos 2\theta = 1 - 2\sin^2$ \theta$$

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

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

Q: Which of the following equations is quadratic in form?

$$2\sin^2$ \theta - \sin \theta - 1 = 0$$

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

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

Q: Which of the following is a correct general solution for $$\tan \theta = 1$$?

$$\theta = \frac{\pi}{4} + k\pi,\ k \in \mathbb{Z}$$

Q: Which of the following is a valid step in solving $$\sin \theta + \cos \theta = 1$$ using identities?

Rewrite $$\sin \theta as$$ $$\sqrt{1 - \cos^2$ \theta}$$

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

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

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

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

Question 1

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

Accepted Answer

$$	heta = \frac{\pi}{3} + 2k\pi,\ 	heta = \frac{5\pi}{3} + 2k\pi,\ k \in \mathbb{Z}$$

Question 2

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

Accepted Answer

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

Question 3

Which identity can be used to rewrite $$1 - 2\sin^2$ 	heta in $$terms of $$\cos 2	heta$$?

Accepted Answer

$$\cos 2	heta = 1 - 2\sin^2$ 	heta$$

Question 4

If $$	an 	heta = -2$$ and $$	heta is in $$the fourth quadrant, which of the following is a solution in $$[0, 2\pi)$$? (Round to two decimal places)

Accepted Answer

$$	heta = 2\pi - 1.11$$

Question 5

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

Accepted Answer

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

Question 6

Which of the following equations is quadratic in form?

Accepted Answer

$$2\sin^2$ 	heta - \sin 	heta - 1 = 0$$

Question 7

A sound wave can be modeled by $$y = 5\sin x + x. If $$the sound level is 3, how many solutions are there for $$x in$$ $$[0, 4\pi]$$?

Accepted Answer

Three

Question 8

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

Accepted Answer

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

Question 9

Which of the following is a correct general solution for $$	an 	heta = 1$$?

Accepted Answer

$$	heta = \frac{\pi}{4} + k\pi,\ k \in \mathbb{Z}$$

Question 10

Given $$\sin 	heta = 2$$, what is the solution set for $$	heta in$$ $$[0, 2\pi)$$?

Accepted Answer

No solution

Question 11

Which of the following is a valid step in solving $$\sin 	heta + \cos 	heta = 1$$ using identities?

Accepted Answer

Rewrite $$\sin 	heta as$$ $$\sqrt{1 - \cos^2$ 	heta}$$

Question 12

If $$\cos 	heta = -1$$, what is the value of $$	heta in$$ $$[0, 2\pi)$$?

Accepted Answer

$$	heta = \pi$$

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

If $$\sin 	heta = \cos 	heta$$, what is the general solution for $$	heta$$?

Accepted Answer

$$	heta = \frac{\pi}{4} + k\pi,\ k \in \mathbb{Z}$$

Question 16

A Ferris wheel rotates such that its height above the ground is modeled by $$h(t) = 10\sin(\omega t) + 12. If $$the height is 12 meters, what is the value of $$\sin(\omega t)$$?

Accepted Answer

$$0$$

Question 17

Which of the following equations has no real solution?

Accepted Answer

$$\sin 	heta = 2$$

Question 18

If $$\sin 	heta = \frac{1}{2}$$, what are the solutions for $$	heta in$$ $$[0, 2\pi)$$?

Accepted Answer

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

Analytic Trigonometry: Solving Trigonometric Equations

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