Trigonometry: Addition Formulas

The addition formulas for the sine and cosine are given by

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

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

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

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

The addition formulas for the tangent are given by

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

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

Ex 1 Find $\sin 75^\circ$ using an addition formula.

Sol $\sin 75^\circ=\sin(45^\circ+30^\circ)=$ $\sin45^\circ\cos30^\circ+\cos45^\circ\sin30^\circ=$ $\frac{\sqrt{2}}{2}\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\frac{1}{2}=$ $\frac{\sqrt{6}}{4}+\frac{\sqrt{2}}{4}=\frac{\sqrt{6}+\sqrt{2}}{4}$.

Ex 2 Find $\cos 15^\circ$ using an addition formula.

Sol $\cos 15^\circ=\cos(45^\circ-30^\circ)=$ $\cos45^\circ\cos30^\circ+\sin45^\circ\sin30^\circ=$ $\frac{\sqrt{2}}{2}\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\frac{1}{2}=$ $\frac{\sqrt{6}}{4}+\frac{\sqrt{2}}{4}=\frac{\sqrt{6}+\sqrt{2}}{4}$.

Pr 1 Find $\cos 75^\circ$ using an addition formula.

Pr 2 Find $\sin 15^\circ$ using an addition formula.

Pr 3 Find $\cos 105^\circ$ using an addition formula.

Pr 4 Use an addition formula to simplify $\sin(\theta+\pi/2)$.

Pr 5 Use an addition formula to simplify $\cos(\pi-\theta)$.

Pr 6 Find $\tan15^\circ$ using an addition formula.

Pr 7 Use the addition formulas to simplify the expression $\frac{1}{2}[\sin(A+B)+\sin(A-B)]$

Pr 8 Find the smallest positive angle between the lines with equations $y=3/2x+8$ and $y=1/5x-6$.
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