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

$$\theta = \dfrac{\pi}{4} + k\pi$$, $$k$$ any integer

Q: Which identity is used to rewrite $$\sin^2$ \theta + \cos \theta = 3 in $$terms of only $$\cos \theta$$?

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

Q: A Ferris wheel rotates such that the angle $$\theta$$ satisfies $$\tan(\theta - \dfrac{\pi}{3}) = 3. $$What are the solutions for $$0 \leq \theta \u003c 2\pi$$?

$$\theta = \dfrac{2\pi}{3}, \dfrac{5\pi}{3}$$

Q: Which of the following is a correct application of the Pythagorean identity to solve $$3\cos^2$ \theta = \sin^2$ \theta$$?

Rewrite $$\sin^2$ \theta as$$ $$1 - \cos^2$ \theta$$ and solve for $$\cos \theta$$

Question 1

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

Accepted Answer

$$	heta = \dfrac{\pi}{4} + k\pi$$, $$k$$ any integer

Question 2

Solve the equation $$2\sin 	heta + 2 = 0$$ for $$0 \leq 	heta \u003c 2\pi. $$Which of the following is the correct solution set?

Accepted Answer

$$\left\{ \dfrac{5\pi}{4}, \dfrac{7\pi}{4} \right\}$$

Question 3

A medical device uses the equation $$	an 	heta = -2 to $$determine the angle of a rotating arm. Using a calculator, which of the following is a solution for $$0 \leq 	heta \u003c 2\pi$$, rounded to two decimal places?

Accepted Answer

$$	heta = 2.03$$

Question 4

Which identity is used to rewrite $$\sin^2$ 	heta + \cos 	heta = 3 in $$terms of only $$\cos 	heta$$?

Accepted Answer

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

Question 5

Solve the equation $$\sin(2	heta) = \dfrac{\sqrt{3}}{2}$$ for $$0 \leq 	heta \u003c 2\pi. $$Which of the following is the correct solution set?

Accepted Answer

$$\left\{ \dfrac{\pi}{6}, \dfrac{\pi}{3}, \dfrac{7\pi}{6}, \dfrac{4\pi}{3} \right\}$$

Question 6

A sound wave is modeled by $$5\sin x + x = 3. $$Using a graphing utility, which of the following is a solution rounded to two decimal places?

Accepted Answer

$$x = 0.52$$

Question 7

Which of the following best describes the process for finding all solutions to $$\cos 	heta = a$$?

Accepted Answer

Find solutions in $$[0, 2\pi)$$ and add multiples of $$2\pi$$

Question 8

Solve the quadratic trigonometric equation $$4\sin^2$ 	heta - 1 = 0$$ for $$0 \leq 	heta \u003c 2\pi. $$Which of the following is the correct solution set?

Accepted Answer

$$\left\{ \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{7\pi}{6}, \dfrac{11\pi}{6} \right\}$$

Question 9

If $$\cos^2$ 	heta - \cos 	heta + 2 = 0$$, what is the solution set for $$0 \leq 	heta \u003c 2\pi$$?

Accepted Answer

The solution set is $$\emptyset (no $$real solutions)

Question 10

A Ferris wheel rotates such that the angle $$	heta$$ satisfies $$	an(	heta - \dfrac{\pi}{3}) = 3. $$What are the solutions for $$0 \leq 	heta \u003c 2\pi$$?

Accepted Answer

$$	heta = \dfrac{2\pi}{3}, \dfrac{5\pi}{3}$$

Question 11

Which of the following is a correct application of the Pythagorean identity to solve $$3\cos^2$ 	heta = \sin^2$ 	heta$$?

Accepted Answer

Rewrite $$\sin^2$ 	heta as$$ $$1 - \cos^2$ 	heta$$ and solve for $$\cos 	heta$$

Question 12

A chemical reaction's rate is modeled by $$Y_1 = 5$$\sin x + x$$ and Y_2$ = 3. $$What does the intersection of these graphs represent?

Accepted Answer

The values of $$x$$ where the reaction rate equals 3

Question 13

Which of the following is the correct general solution for $$\sin 	heta = -\dfrac{1}{2}$$?

Accepted Answer

$$	heta = \dfrac{7\pi}{6} + 2k\pi or$$ $$	heta = \dfrac{11\pi}{6} + 2k\pi$$, $$k$$ any integer

Question 14

If $$\sin 	heta = a$$ and $$-1 \leq a \leq 1$$, how many solutions are there in $$[0, 2\pi)$$?

Accepted Answer

Two solutions, unless $$a = 1 or$$ $$a = -1$$

Analytic Trigonometry: Solving Trigonometric Equations

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