$$\tan(75^\circ) = \dfrac{\tan 45^\circ + \tan 30^\circ}{1 - \tan 45^\circ \tan 30^\circ}$$

$$\tan(A+B) = \dfrac{\tan A + \tan B}{1 - \tan A \tan B}$$

Question 1

Which of the following is the correct formula for the cosine of the sum of two angles?

Accepted Answer

$$\cos(A+B) = \cos A \cos B - \sin A \sin B$$

Question 2

Given $$\sin x = \dfrac{3}{5}$$ and $$x is in $$Quadrant II, what is $$\cos x$$?

Accepted Answer

$$-\dfrac{4}{5}$$

Question 3

If $$	an 	heta = -2$$ and $$	heta is in $$Quadrant II, what is $$\sin 	heta$$?

Accepted Answer

$$\dfrac{2}{\sqrt{5}}$$

Question 4

Which identity is used to simplify $$\sin^2$ x + \cos^2$ x$$?

Accepted Answer

$$\sin^2$ x + \cos^2$ x = 1$$

Question 5

Simplify $$	an(75^\circ)$$ using the sum formula for tangent.

Accepted Answer

$$	an(75^\circ) = \dfrac{	an 45^\circ + 	an 30^\circ}{1 - 	an 45^\circ 	an 30^\circ}$$

Question 6

If $$\sin \alpha = \dfrac{5}{13}$$ and $$\alpha is in $$Quadrant II, find $$\cos \alpha.$$

Accepted Answer

$$-\dfrac{12}{13}$$

Question 7

Which of the following is the correct double angle formula for sine?

Accepted Answer

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

Question 8

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

Accepted Answer

$$\cos^2$ x = \dfrac{1 + \cos(2x)}{2}$$

Question 9

If $$\sin x = \dfrac{1}{2}$$ and $$x is in $$Quadrant I, what is $$x in $$degrees?

Accepted Answer

$$30^\circ$$

Question 10

Find the exact value of $$\sin\left(\dfrac{5\pi}{12}\right)$$ using sum or difference identities.

Accepted Answer

$$\sin\left(\dfrac{5\pi}{12}\right) = \sin\left(\dfrac{\pi}{4} + \dfrac{\pi}{6}\right)$$

Question 11

If $$	an x = 3$$ and $$x is in $$Quadrant III, what is $$\sin x$$?

Accepted Answer

$$-\dfrac{3}{\sqrt{10}}$$

Question 12

Which of the following is the correct half-angle formula for sine?

Accepted Answer

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

Question 13

Given $$\cos(2x) = -\dfrac{1}{2}$$ and $$x is in $$Quadrant II, what is $$\sin x$$?

Accepted Answer

$$\dfrac{\sqrt{3}}{2}$$

Question 14

If $$\sin x = -\dfrac{3}{5}$$ and $$x is in $$Quadrant IV, what is $$\cos x$$?

Accepted Answer

$$\dfrac{4}{5}$$

Question 15

Which of the following is the correct formula for $$	an(A+B)$$?

Accepted Answer

$$	an(A+B) = \dfrac{	an A + 	an B}{1 - 	an A 	an B}$$

Question 16

If $$\cos(2x) = 2\cos^2$ x - 1$$, what is $$\cos^2$ x$$?

Accepted Answer

$$\cos^2$ x = \dfrac{1 + \cos(2x)}{2}$$

Question 17

Find the exact value of $$\sin\left(\dfrac{15^\circ}{2}\right)$$ using the half-angle identity.

Accepted Answer

$$\sin\left(\dfrac{15^\circ}{2}\right) = \sqrt{\dfrac{1 - \cos 15^\circ}{2}}$$

Question 18

If $$\sin x = \dfrac{1}{3}$$ and $$x is in $$Quadrant II, what is $$\cos x$$?

Accepted Answer

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

Trigonometric Identities, Equations, and Applications: Study Guide for Exam 3

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