Q: Which of the following is the correct double-angle formula for $$\sin(2\theta)$$?

$$\sin(2\theta) = 2\sin \theta \cos \theta$$

Q: Which of the following is an equivalent expression for $$\cos^4$ \theta$$ that does not involve powers greater than 1?

$$\frac{3}{8} + \frac{1}{2}\cos(2\theta) + \frac{1}{8}\cos(4\theta)$$

Q: Which formula expresses $$\tan(2\theta) in $$terms of $$\tan \theta$$?

$$\tan(2\theta) = \frac{2\tan \theta}{1 - \tan^2$ \theta}$$

Q: Solve for $$\theta in$$ $$\sin \theta \cos \theta = -\frac{1}{4}$$ for $$0 \leq \theta \u003c 2\pi.$$

$$\theta = \frac{7\pi}{12}, \frac{11\pi}{12}, \frac{19\pi}{12}, \frac{23\pi}{12}$$

Q: If $$\cos \theta = 0.6$$, what is $$\sin^2$ \theta$$ using a double-angle identity?

$$\frac{1 - \cos(2\theta)}{2}$$

Q: Which of the following is the correct double-angle formula for $$\cos(2\theta)$$?

$$\cos(2\theta) = 2\cos^2$ \theta - 1$$

Question 1

Which of the following is the correct double-angle formula for $$\sin(2	heta)$$?

Accepted Answer

$$\sin(2	heta) = 2\sin 	heta \cos 	heta$$

Question 2

Given $$\sin 	heta = \frac{5}{6}$$ and $$\frac{\pi}{2} \u003c 	heta \u003c \pi$$, what is $$\cos 	heta$$?

Accepted Answer

$$-\frac{\sqrt{11}}{6}$$

Question 3

If $$\sin 	heta = \frac{5}{6}$$ and $$\frac{\pi}{2} \u003c 	heta \u003c \pi$$, find $$\sin(2	heta).$$

Accepted Answer

$$-\frac{5\sqrt{11}}{18}$$

Question 4

Which of the following is an equivalent expression for $$\cos^4$ 	heta$$ that does not involve powers greater than 1?

Accepted Answer

$$\frac{3}{8} + \frac{1}{2}\cos(2	heta) + \frac{1}{8}\cos(4	heta)$$

Question 5

Which formula expresses $$	an(2	heta) in $$terms of $$	an 	heta$$?

Accepted Answer

$$	an(2	heta) = \frac{2	an 	heta}{1 - 	an^2$ 	heta}$$

Question 6

A projectile is launched with initial velocity $$v_0 at an $$angle $$	heta to $$the horizontal. The range is $$R = \frac{v_0$^2 \sin(2	heta)}{32}. $$For a fixed $$v_0$$, at what angle $$	heta is $$the range maximized?

Accepted Answer

$$45^\circ$$

Question 7

Use a half-angle formula to find $$\cos 75^\circ.$$

Accepted Answer

$$\frac{\sqrt{2 - \sqrt{3}}}{2}$$

Question 8

If $$\cos \alpha = -\frac{2}{3}$$ and $$\pi \u003c \alpha \u003c \frac{3\pi}{2}$$, what is $$\sin \frac{\alpha}{2}$$?

Accepted Answer

$$\frac{\sqrt{5}}{6}$$

Question 9

Given $$\cos \alpha = -\frac{2}{3}$$ and $$\pi \u003c \alpha \u003c \frac{3\pi}{2}$$, what is $$\cos \frac{\alpha}{2}$$?

Accepted Answer

$$-\frac{1}{6}$$

Question 10

Solve for $$	heta in$$ $$\sin 	heta \cos 	heta = -\frac{1}{4}$$ for $$0 \leq 	heta \u003c 2\pi.$$

Accepted Answer

$$	heta = \frac{7\pi}{12}, \frac{11\pi}{12}, \frac{19\pi}{12}, \frac{23\pi}{12}$$

Question 11

Which of the following is the correct half-angle formula for $$	an \frac{\alpha}{2}$$?

Accepted Answer

All of the above

Question 12

If $$	an 	heta = 3$$, what is $$	an(2	heta)$$?

Accepted Answer

$$-\frac{6}{8}$$

Question 13

Which of the following is the correct half-angle formula for $$\sin \frac{\alpha}{2}$$?

Accepted Answer

$$\sin \frac{\alpha}{2} = \pm \sqrt{\frac{1 - \cos \alpha}{2}}$$

Question 14

If $$\cos 	heta = 0.6$$, what is $$\sin^2$ 	heta$$ using a double-angle identity?

Accepted Answer

$$\frac{1 - \cos(2	heta)}{2}$$

Question 15

Which of the following is the correct double-angle formula for $$\cos(2	heta)$$?

Accepted Answer

$$\cos(2	heta) = 2\cos^2$ 	heta - 1$$

Analytic Trigonometry: Double-Angle and Half-Angle Formulas

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