Math Core

This test covers the fundamental identities, simplifying and verifying identities, the sum, difference and double-angle formulas, and solving trigonometric equations.

Question 1

If $\cos\theta = -\dfrac{5}{13}$ and $\theta$ is in Quadrant III, find $\tan\theta$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 2

Simplify $\csc x - \sin x$ .

Question 3

Which equation is not an identity?

Question 4

If $\sin\theta - \cos\theta = \dfrac{1}{3}$ , find $\sin 2\theta$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 5

Find the exact value of $\sin\dfrac{\pi}{12}$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 6

Find the exact value of $\cos 105^\circ$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 7

If $\tan\alpha = 3$ and $\tan\beta = \dfrac{1}{2}$ , find $\tan(\alpha - \beta)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 8

If $\sin\theta = \dfrac{5}{13}$ and $\theta$ is in Quadrant II, find $\cos 2\theta$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 9

Simplify $\cos^4 x - \sin^4 x$ .

Question 10

Find the exact value of $\sin 15^\circ\cos 15^\circ$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Question 11

Solve $2\cos x - \sqrt3 = 0$ on $[0, 2\pi)$ .

Separate answers with commas, e.g. 2, -5

Question 12

Solve $2\sin^2 x - \sin x - 1 = 0$ on $[0, 2\pi)$ .

Separate answers with commas, e.g. 2, -5

Question 13

Solve $\cos 2x = \cos x$ on $[0, 2\pi)$ .

Separate answers with commas, e.g. 2, -5

Question 14

Solve $\tan 2x = 1$ on $[0, 2\pi)$ .

Separate answers with commas, e.g. 2, -5

Question 15

Which gives all solutions of $\sin x = \dfrac{\sqrt3}{2}$ ? ( $k$ is any integer.)