Q: Which of the following is the general solution to the equation $$\sin \theta = \dfrac{1}{2}$$, where $$\theta is in $$radians and $$k is $$any integer?

$$\theta = \dfrac{\pi}{6} + 2k\pi or$$ $$\theta = \dfrac{5\pi}{6} + 2k\pi$$

Q: Find all solutions to $$\tan \theta + 1 = 0 on $$the interval $$[0, 2\pi). $$Select the correct answer.

$$\theta = \dfrac{3\pi}{4}, \dfrac{7\pi}{4}$$

Q: Use the half-angle formula to evaluate $$\tan\left(\dfrac{3\pi}{8}\right). $$Which formula and sign should be used, and what is the exact value?

$$\tan\left(\dfrac{3\pi}{8}\right) = 1 + \sqrt{2}$$

Q: Solve $$2\sin^2$\theta - 3\sin\theta + 1 = 0$$ for $$\theta in$$ $$[0, 2\pi). $$What are the solutions?

$$\theta = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{\pi}{2}$$

Q: Suppose $$\cos \theta = \dfrac{4}{5}$$ and $$\theta is in $$Quadrant IV. What is $$\tan 2\theta$$?

$$\tan 2\theta = -\dfrac{24}{7}$$

Q: Which sequence of steps verifies the identity $$\tan x + 1 \over \sec x = \cos x + \sin x$$?

$$\dfrac{\tan x + 1}{\sec x} = \dfrac{\sin x}{\cos x} + 1 \over \dfrac{1}{\cos x} = \cos x + \sin x$$

Q: Solve $$\cos 2\theta - 1 = 0$$ for $$\theta in$$ $$[0, 2\pi). $$What are the solutions?

$$\theta = \dfrac{\pi}{8}, \dfrac{7\pi}{8}, \dfrac{9\pi}{8}, \dfrac{15\pi}{8}$$

Q: Which of the following verifies the identity $$\tan \theta \cdot \cos \theta \cdot \csc \theta = 1$$?

$$\tan \theta \cdot \cos \theta \cdot \csc \theta = \dfrac{\sin \theta}{\cos \theta} \cdot \cos \theta \cdot \dfrac{1}{\sin \theta} = 1$$

Question 1

Which of the following is the general solution to the equation $$\sin 	heta = \dfrac{1}{2}$$, where $$	heta is in $$radians and $$k is $$any integer?

Accepted Answer

$$	heta = \dfrac{\pi}{6} + 2k\pi or$$ $$	heta = \dfrac{5\pi}{6} + 2k\pi$$

Question 2

Find all solutions to $$	an 	heta + 1 = 0 on $$the interval $$[0, 2\pi). $$Select the correct answer.

Accepted Answer

$$	heta = \dfrac{3\pi}{4}, \dfrac{7\pi}{4}$$

Question 3

Rewrite $$2	an\left(\dfrac{5\pi}{8}ight) \big/ \left(1 - 	an^2$\left(\dfrac{5\pi}{8}ight)ight) as a $$single trigonometric function and find its exact value.

Accepted Answer

$$1$$

Question 4

Suppose $$\sin \alpha = \dfrac{12}{13}$$ with $$\alpha in $$Quadrant I, and $$\cos \beta = \dfrac{3}{11}$$ with $$\beta in $$Quadrant IV. What is $$\sin(\alpha + \beta)$$?

Accepted Answer

$$\dfrac{12}{13} \cdot \dfrac{3}{11} + \dfrac{5}{13} \cdot \left(-\dfrac{2}{11}\right)$$

Question 5

Use the half-angle formula to evaluate $$	an\left(\dfrac{3\pi}{8}ight). $$Which formula and sign should be used, and what is the exact value?

Accepted Answer

$$	an\left(\dfrac{3\pi}{8}ight) = 1 + \sqrt{2}$$

Question 6

Solve $$2\sin^2$	heta - 3\sin	heta + 1 = 0$$ for $$	heta in$$ $$[0, 2\pi). $$What are the solutions?

Accepted Answer

$$	heta = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{\pi}{2}$$

Question 7

Suppose $$\cos 	heta = \dfrac{4}{5}$$ and $$	heta is in $$Quadrant IV. What is $$	an 2	heta$$?

Accepted Answer

$$	an 2	heta = -\dfrac{24}{7}$$

Question 8

Find the exact value of $$\sin\left(\dfrac{7\pi}{6} - \dfrac{3\pi}{4}\right).$$

Accepted Answer

$$\dfrac{1 + \sqrt{3}}{2\sqrt{2}}$$

Question 9

A Ferris wheel rotates such that the height $$h (in $$meters) of a passenger above the ground at time $$t (in $$seconds) is modeled by $$h(t) = 10 + 8\sin\left(\dfrac{\pi}{15}t\right). At $$what time $$t in$$ $$[0, 30) is $$the passenger at a height of $$18$$ meters?

Accepted Answer

$$t = 7.5$$ seconds

Question 10

Which sequence of steps verifies the identity $$	an x + 1 \over \sec x = \cos x + \sin x$$?

Accepted Answer

$$\dfrac{	an x + 1}{\sec x} = \dfrac{\sin x}{\cos x} + 1 \over \dfrac{1}{\cos x} = \cos x + \sin x$$

Question 11

Use the double-angle formula to find $$\sin\left(2	an$$^{-1}$$\left(\dfrac{2}{7}ight)ight). $$What is the exact value?

Accepted Answer

$$\dfrac{28}{53}$$

Question 12

Solve $$\cos 2	heta - 1 = 0$$ for $$	heta in$$ $$[0, 2\pi). $$What are the solutions?

Accepted Answer

$$	heta = \dfrac{\pi}{8}, \dfrac{7\pi}{8}, \dfrac{9\pi}{8}, \dfrac{15\pi}{8}$$

Question 13

Suppose $$	an \alpha = \dfrac{4}{3}$$ and $$\pi \u003c \alpha \u003c \dfrac{3\pi}{2}. In $$which quadrant does $$\dfrac{\alpha}{2}$$ lie, and what is the sign of $$	an\left(\dfrac{\alpha}{2}ight)$$?

Accepted Answer

Quadrant II, negative

Question 14

Which of the following verifies the identity $$	an 	heta \cdot \cos 	heta \cdot \csc 	heta = 1$$?

Accepted Answer

$$	an 	heta \cdot \cos 	heta \cdot \csc 	heta = \dfrac{\sin 	heta}{\cos 	heta} \cdot \cos 	heta \cdot \dfrac{1}{\sin 	heta} = 1$$

Question 15

Find the exact value of $$\sin\left(\dfrac{5\pi}{12}\right)\cos\left(\dfrac{\pi}{3}\right) + \cos\left(\dfrac{5\pi}{12}\right)\sin\left(\dfrac{\pi}{3}\right).$$

Accepted Answer

$$\sin\left(\dfrac{5\pi}{12} + \dfrac{\pi}{3}\right) = \sin\left(\dfrac{9\pi}{12}\right) = \sin\left(\dfrac{3\pi}{4}\right) = \dfrac{1}{2}$$

Question 16

Solve $$\cos 2	heta - 1 = -3\cos 	heta$$ for $$	heta in$$ $$[0, 2\pi). $$What are the solutions?

Accepted Answer

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

Question 17

Suppose $$\cos 	heta = \dfrac{24}{25}$$ and $$	heta is in $$Quadrant IV. What is $$\sin 2	heta$$?

Accepted Answer

$$\sin 2	heta = -\dfrac{336}{625}$$

Question 18

Which sequence of steps verifies the identity $$\dfrac{\sec 2x - 	an 2x}{\sec x + 	an x} = \sec x - 	an x$$?

Accepted Answer

$$\dfrac{\sec 2x - 	an 2x}{\sec x + 	an x} = \dfrac{(\sec x + 	an x)(\sec x - 	an x)}{\sec x + 	an x} = \sec x - 	an x$$

Precalculus Practice Test 3 (Trigonometric Equations, Identities, and Formulas) – Step-by-Step Guidance

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