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

$$\pm \sqrt{\frac{1 - \cos \theta}{2}}$$

Q: Given $$\cos(2\theta) = 2\cos^2$\theta - 1$$, what is $$\cos^2$\theta in $$terms of $$\cos(2\theta)$$?

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

Q: Which of the following is the correct half-angle formula for $$\tan \frac{\theta}{2}$$?

$$\pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}$$

Q: Which of the following is a correct identity for $$\sin^2$ \theta$$ using a double-angle formula?

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

Q: A medical imaging device uses the formula $$I = I_0$ \cos^2$ \theta$$ for intensity. If the angle is halved, what is the new intensity in terms of $$I_0$$ and $$\cos \theta$$?

$$I_0$$ \frac{1 + \cos \theta}{2}$$

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

$$\frac{2\tan \theta}{1 - \tan^2$ \theta}$$

Question 1

Given $$	an 	heta = \frac{3}{2}$$ and $$\frac{\pi}{2} \u003c 	heta \u003c \pi$$, what is $$\sin(2	heta)$$?

Accepted Answer

$$-\frac{12}{13}$$

Question 2

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

Accepted Answer

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

Question 3

If $$\sin 	heta = -\frac{6}{\sqrt{61}}$$ and $$\cos 	heta = -\frac{5}{\sqrt{61}}$$, what is $$	an(2	heta)$$?

Accepted Answer

$$-\frac{60}{11}$$

Question 4

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

Accepted Answer

$$\pm \sqrt{\frac{1 - \cos 	heta}{2}}$$

Question 5

Find the exact value of $$\sin\left(\frac{5\pi}{12}\right)$$ using the half-angle formula.

Accepted Answer

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

Question 6

A rain gutter is to be constructed from a 15-inch wide aluminum sheet by bending up 5 inches on each side at an angle $$	heta. $$The area of the opening is $$A(	heta) = 25 \sin 	heta (\cos 	heta + 1). $$What value of $$	heta$$ maximizes the area?

Accepted Answer

$$45^\circ$$

Question 7

Given $$\cos(2	heta) = 2\cos^2$	heta - 1$$, what is $$\cos^2$	heta in $$terms of $$\cos(2	heta)$$?

Accepted Answer

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

Question 8

If $$	an 	heta = 2$$ and $$0 \u003c 	heta \u003c \frac{\pi}{2}$$, what is $$	an(2	heta)$$?

Accepted Answer

$$\frac{4}{3}$$

Question 9

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

Accepted Answer

$$\pm \sqrt{\frac{1 - \cos 	heta}{1 + \cos 	heta}}$$

Question 10

Solve for $$	heta in$$ $$2\cos(2	heta) + 10\sin 	heta = 5$$ for $$0 \leq 	heta \u003c 2\pi.$$

Accepted Answer

$$\frac{\pi}{4}, \frac{3\pi}{4}$$

Question 11

If $$\sin 	heta = \frac{1}{3}$$ and $$0 \u003c 	heta \u003c \frac{\pi}{2}$$, what is $$\cos(2	heta)$$?

Accepted Answer

$$\frac{5}{9}$$

Question 12

Which of the following is a correct identity for $$\sin^2$ 	heta$$ using a double-angle formula?

Accepted Answer

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

Question 13

Given $$\sin 	heta = \frac{3}{5}$$ and $$	heta is in $$the first quadrant, what is $$\cos(2	heta)$$?

Accepted Answer

$$\frac{7}{25}$$

Question 14

A medical imaging device uses the formula $$I = I_0$ \cos^2$ 	heta$$ for intensity. If the angle is halved, what is the new intensity in terms of $$I_0$$ and $$\cos 	heta$$?

Accepted Answer

$$I_0$$ \frac{1 + \cos 	heta}{2}$$

Question 15

If $$\sin(2	heta) = 0.8$$ and $$	heta is in $$the first quadrant, what is $$\sin 	heta$$?

Accepted Answer

$$0.6$$

Question 16

Which of the following is a correct application of the double-angle formula for $$	an(2	heta)$$?

Accepted Answer

$$\frac{2	an 	heta}{1 - 	an^2$ 	heta}$$

Question 17

If $$\cos 	heta = -\frac{5}{\sqrt{61}}$$ and $$\sin 	heta = -\frac{6}{\sqrt{61}}$$, what is $$\cos(2	heta)$$?

Accepted Answer

$$-\frac{11}{61}$$

Question 18

A satellite dish is designed so that the angle of elevation $$	heta$$ maximizes the signal area $$A(	heta) = 25 \sin 	heta (\cos 	heta + 1). $$What is the maximum area?

Accepted Answer

$$25(\sqrt{2} + 1)$$

Question 19

If $$\sin 	heta = \frac{1}{2}$$ and $$0 \u003c 	heta \u003c \frac{\pi}{2}$$, what is $$\sin^2$ \frac{	heta}{2}$$?

Accepted Answer

$$\frac{1 - \sqrt{3}/2}{2}$$

Study Guide: Double-Angle and Half-Angle Formulas in Precalculus

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