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 identities expresses $$\cos(2\theta) in $$terms of $$\sin\theta$$ only?

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

Q: Which of the following is a correct application of the double-angle formula for cosine?

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

Q: Which of the following is NOT a correct double-angle identity?

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

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 $$\cos	heta = -\frac{1}{6}$$ and $$	heta is in $$quadrant II, what is $$\sin(2	heta)$$?

Accepted Answer

$$-\frac{5\sqrt{35}}{18}$$

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

$$45^\circ$$

Question 5

If $$	an	heta = -2$$ and $$	heta is in $$quadrant II, what is $$	an(2	heta)$$?

Accepted Answer

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

Question 6

Which of the following identities expresses $$\cos(2	heta) in $$terms of $$\sin	heta$$ only?

Accepted Answer

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

Question 7

Given $$\sin\alpha = \frac{3}{5}$$ and $$\alpha is in $$quadrant II, find $$\cos\left(\frac{\alpha}{2}\right).$$

Accepted Answer

Question 8

Which of the following is a correct application of the double-angle formula for cosine?

Accepted Answer

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

Question 9

If $$\sin	heta = \frac{5}{6}$$ and $$	heta is in $$quadrant II, what is $$\cos(2	heta)$$?

Accepted Answer

$$-\frac{7}{18}$$

Question 10

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

Accepted Answer

All of the above

Question 11

A satellite is launched at an angle $$	heta to $$the horizontal. If the range is given by $$R = \frac{v_0$^2 \sin(2	heta)}{g}$$, what is the physical significance of the double-angle in this context?

Accepted Answer

It determines the horizontal distance traveled as a function of the launch angle.

Question 12

If $$\cos\alpha = -\frac{3}{5}$$ and $$\alpha is in $$quadrant III, what is $$\sin\left(\frac{\alpha}{2}\right)$$?

Accepted Answer

$$-\frac{1}{2}$$

Question 13

Which of the following is NOT a correct double-angle identity?

Accepted Answer

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

Question 14

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

Accepted Answer

$$-\frac{24}{7}$$

Question 15

Which of the following expressions is equivalent to $$1 - 2\sin^2$	heta$$?

Accepted Answer

$$\cos(2	heta)$$

Question 16

If $$\cos	heta = \frac{4}{5}$$ and $$	heta is in $$quadrant IV, what is $$\sin\left(\frac{	heta}{2}ight)$$?

Accepted Answer

$$\frac{1}{3}$$

Analytic Trigonometry: Double-Angle and Half-Angle Formulas

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