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 the correct power-reducing formula for $$\sin^2$\theta$$?

$$\sin^2$\theta = \frac{1 - \cos 2\theta}{2}$$

Q: Which half-angle formula would you use to compute $$\sin(22.5^\circ)$$?

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

Q: Rewrite $$\cos^4$\theta in $$terms of first powers of cosine using power-reducing formulas.

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

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

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

Q: Which of the following is the correct power-reducing formula for $$\tan^2$\theta$$?

$$\tan^2$\theta = \frac{1 - \cos 2\theta}{1 + \cos 2\theta}$$

Q: Which of the following is the correct half-angle formula for $$\tan\frac{\theta}{2}$$ that does NOT involve a plus or minus?

$$\tan\frac{\theta}{2} = \frac{\sin\theta}{1 + \cos\theta}$$

Q: Rewrite $$\sin^2$\theta\cos^2$\theta in $$terms of a double angle.

$$\frac{1}{8}(1 - \cos 4\theta)$$

Q: Which of the following identities is used to establish $$\cos 2\theta = 2\cos^2$\theta - 1$$?

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

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 $$\cos	heta = -0.8$$ and $$90^\circ \leq 	heta \leq 180^\circ$$, what is $$\sin(2	heta)$$?

Accepted Answer

$$0.96$$

Question 3

Which of the following is the correct power-reducing formula for $$\sin^2$	heta$$?

Accepted Answer

$$\sin^2$	heta = \frac{1 - \cos 2	heta}{2}$$

Question 4

If $$\cot	heta = 2$$ and $$\sin	heta \u003c 0$$, what is $$	an(2	heta)$$?

Accepted Answer

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

Question 5

Which half-angle formula would you use to compute $$\sin(22.5^\circ)$$?

Accepted Answer

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

Question 6

Rewrite $$\cos^4$	heta in $$terms of first powers of cosine using power-reducing formulas.

Accepted Answer

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

Question 7

Given $$\cos	heta = -0.8$$ and $$90^\circ \leq 	heta \leq 180^\circ$$, what is $$	an\left(\frac{	heta}{2}ight)$$?

Accepted Answer

$$0.3$$

Question 8

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

Accepted Answer

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

Question 9

Evaluate $$\cos(165^\circ)$$ using a half-angle formula.

Accepted Answer

$$-\frac{\sqrt{6} + \sqrt{2}}{4}$$

Question 10

Which of the following is the correct power-reducing formula for $$	an^2$	heta$$?

Accepted Answer

$$	an^2$	heta = \frac{1 - \cos 2	heta}{1 + \cos 2	heta}$$

Question 11

If $$\sin	heta = -\frac{2}{\sqrt{5}}$$ and $$\cos	heta = -\frac{1}{\sqrt{5}}$$, what is $$\sin(2	heta)$$?

Accepted Answer

$$\frac{4}{5}$$

Question 12

Which of the following is the correct half-angle formula for $$	an\frac{	heta}{2}$$ that does NOT involve a plus or minus?

Accepted Answer

$$	an\frac{	heta}{2} = \frac{\sin	heta}{1 + \cos	heta}$$

Question 13

Rewrite $$\sin^2$	heta\cos^2$	heta in $$terms of a double angle.

Accepted Answer

$$\frac{1}{8}(1 - \cos 4	heta)$$

Question 14

Which of the following identities is used to establish $$\cos 2	heta = 2\cos^2$	heta - 1$$?

Accepted Answer

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

Question 15

A satellite dish is designed so that the angle between two support beams is $$2	heta. If$$ $$\sin	heta = 0.6$$ and $$	heta is in $$the second quadrant, what is $$\cos(2	heta)$$?

Accepted Answer

$$-0.28$$

Question 16

Evaluate $$	an(22.5^\circ)$$ using a half-angle formula.

Accepted Answer

$$\sqrt{2} - 1$$

Question 17

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

Accepted Answer

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

Question 18

Rewrite $$\sin^4$	heta in $$terms of first powers of cosine using power-reducing formulas.

Accepted Answer

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

Double-Angle, Power-Reducing, and Half-Angle Formulas in Trigonometry 5.3

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