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

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

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

$$\sin\left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \cos \theta}{2}}$$

Q: Which expression is equivalent to $$\cos^2$ \theta$$ using a double-angle formula?

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

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

$$\tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos \theta}{\sin \theta}$$

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

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

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

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

Q: Which expression is equivalent to $$\sin^2$ \theta$$ using a double-angle formula?

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

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 $$\sin 	heta = \frac{3}{5}$$ and $$0 \u003c 	heta \u003c \frac{\pi}{2}$$, what is the exact value of $$\cos(2	heta)$$?

Accepted Answer

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

Question 3

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 4

If $$\cos \alpha = \frac{3}{5}$$ and $$0 \u003c \alpha \u003c \frac{\pi}{2}$$, what is the exact value of $$\sin(2\alpha)$$?

Accepted Answer

$$\frac{24}{25}$$

Question 5

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

Accepted Answer

$$\sin\left(\frac{	heta}{2}ight) = \sqrt{\frac{1 - \cos 	heta}{2}}$$

Question 6

If $$\cos 	heta = \frac{1}{2}$$ and $$0 \u003c 	heta \u003c \pi$$, what is the exact value of $$\sin\left(\frac{	heta}{2}ight)$$?

Accepted Answer

$$\frac{1}{\sqrt{2}}$$

Question 7

Which expression is equivalent to $$\cos^2$ 	heta$$ using a double-angle formula?

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

$$	an\left(\frac{	heta}{2}ight) = \frac{1 - \cos 	heta}{\sin 	heta}$$

Question 10

If $$\cos \alpha = \frac{3}{5}$$ and $$0 \u003c \alpha \u003c \frac{\pi}{2}$$, what is the exact value of $$	an\left(\frac{\alpha}{2}ight)$$?

Accepted Answer

$$\frac{1}{4}$$

Question 11

An object is propelled upward at an angle $$	heta to $$the horizontal with initial velocity $$v_0. $$Ignoring air resistance, the range $$R is $$given by $$R(	heta) = \frac{v_0$^2 \sin(2	heta)}{g}. $$For what angle $$	heta is $$the range maximized?

Accepted Answer

$$	heta = 45^\circ$$

Question 12

If $$\cos 	heta = -\frac{1}{2}$$ and $$\pi \u003c 	heta \u003c 2\pi$$, what is the exact value of $$\sin\left(\frac{	heta}{2}ight)$$?

Accepted Answer

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

Question 13

Which formula expresses $$\cos(2	heta) in $$terms of $$\sin 	heta$$?

Accepted Answer

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

Question 14

If $$\sin 	heta = \frac{4}{5}$$ and $$0 \u003c 	heta \u003c \frac{\pi}{2}$$, what is the exact value of $$\cos(2	heta)$$?

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

If $$\cos 	heta = \frac{3}{5}$$ and $$0 \u003c 	heta \u003c \frac{\pi}{2}$$, what is the exact value of $$\cos\left(\frac{	heta}{2}ight)$$?

Accepted Answer

$$\frac{2}{\sqrt{8}}$$

Question 17

Which expression is equivalent to $$\sin^2$ 	heta$$ using a double-angle formula?

Accepted Answer

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

Question 18

If $$\cos 	heta = \frac{1}{2}$$ and $$0 \u003c 	heta \u003c \pi$$, what is the exact value of $$	an\left(\frac{	heta}{2}ight)$$?

Accepted Answer

$$\sqrt{3}$$

Double-Angle and Half-Angle Formulas in Analytic Trigonometry

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