Sum and Difference Identities

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Show your work for each problem.

1.
Which expression is equal to $\sin(A + B)$ ?
1. a) $\sin A \sin B + \cos A \cos B$
2. b) $\cos A \cos B - \sin A \sin B$
3. c) $\sin A \cos B + \cos A \sin B$
4. d) $\sin A + \sin B$
2.
Simplify without a calculator: $\sin 70^\circ \cos 20^\circ + \cos 70^\circ \sin 20^\circ$
3.
Use the cosine difference identity to find $\cos(90^\circ - 30^\circ)$ .
4.
Which expression is equal to $\cos(A - B)$ ?
1. a) $\sin A \cos B - \cos A \sin B$
2. b) $\cos A \cos B - \sin A \sin B$
3. c) $\cos A \cos B + \sin A \sin B$
4. d) $\cos A - \cos B$
5.
Simplify without a calculator: $\cos 50^\circ \cos 10^\circ - \sin 50^\circ \sin 10^\circ$
6.
Find $\sin 105^\circ$ with $105^\circ = 60^\circ + 45^\circ$ .
7.
What is the exact value of $\sin 75^\circ$ ?
1. a) $\frac{\sqrt{3} + 1}{4}$
2. b) $\frac{1}{2} + \frac{\sqrt{2}}{2}$
3. c) $\frac{\sqrt{6} - \sqrt{2}}{4}$
4. d) $\frac{\sqrt{6} + \sqrt{2}}{4}$
8.
Use $\tan(45^\circ - 30^\circ)$ with $\tan 45^\circ = 1$ and $\tan 30^\circ = \frac{\sqrt{3}}{3}$ to find $\tan 15^\circ$ . Round to 4 decimal places.
9.
$\sin A = \frac{5}{13}$ with $A$ in Quadrant I, and $\cos B = -\frac{3}{5}$ with $B$ in Quadrant II. Find $\sin(A + B)$ .
10.
Find $\cos 75^\circ$ with $75^\circ = 45^\circ + 30^\circ$ .
11.
Which value is equal to $\cos 80^\circ \cos 20^\circ + \sin 80^\circ \sin 20^\circ$ ?
1. a) $\cos 60^\circ$
2. b) $\sin 100^\circ$
3. c) $\cos 100^\circ$
4. d) $\sin 60^\circ$
12.
$\sin A = \frac{4}{5}$ and $\cos B = \frac{12}{13}$ , with $A$ and $B$ in Quadrant I. Find $\cos(A + B)$ as a fraction.
13.
$\tan A = 2$ and $\tan B = 3$ , with $A$ and $B$ acute angles. Find $\tan(A + B)$ and the angle $A + B$ .
14.
$\sin A = \frac{3}{5}$ with $A$ in Quadrant I. Use $\cos(A + B)$ with $B = A$ to find $\cos 2A$ .
15.
What is the exact value of $\cos 165^\circ$ ? (Hint: $165^\circ = 120^\circ + 45^\circ$ )
1. a) $\frac{\sqrt{6} - \sqrt{2}}{4}$
2. b) $\frac{\sqrt{6} + \sqrt{2}}{4}$
3. c) $\frac{\sqrt{2} - \sqrt{6}}{4}$
4. d) $-\frac{\sqrt{6} + \sqrt{2}}{4}$
16.
$\cos A = -\frac{3}{5}$ with $A$ in Quadrant II, and $\sin B = -\frac{5}{13}$ with $B$ in Quadrant III. Find $\sin(A - B)$ as a fraction.
17.
$\cos A = -\frac{4}{5}$ with $A$ in Quadrant III, and $\sin B = \frac{12}{13}$ with $B$ in Quadrant II. Find $\cos(A + B)$ .
18.
Find $\tan 105^\circ$ with $105^\circ = 60^\circ + 45^\circ$ .