Trigonometry: Double-Angle and Half-Angle Formulas

Double-Angle Formulas

$\sin 2\theta=2\sin\theta \cos\theta$

$\cos 2\theta=\cos^2 \theta-\sin^2 \theta$2 mm $\cos 2\theta=2\cos^2 \theta-1$2 mm $\cos 2\theta=1-2\sin^2 \theta$

Half-Angle Formulas

$\cos^2 \theta=\frac{1}{2}(1+\cos 2\theta)$, so

$$\cos\theta/2=\pm\sqrt{\frac{1+\cos\theta}{2}}$$

$\sin^2 \theta=\frac{1}{2}(1-\cos 2\theta)$, so

$$\sin\theta/2=\pm\sqrt{\frac{1-\cos\theta}{2}}$$

Ex 1 Find $\sin 15^\circ$ using a half-angle formula.

Sol

$$\sin 15^\circ=+\sqrt{\frac{1-\cos 30^\circ}{2}}$$

$$=\sqrt{\frac{1-\frac{\sqrt{3}}{2}}{2}}$$

$$=\sqrt{\frac{2-\sqrt{3}}{4}}=\frac{\sqrt{2-\sqrt{3}}}{2}$$

Ex 2 Find $\sin 2\theta$ if $\sin\theta=3/5$ and $\theta$ is in Quadrant II.

Sol We have that $\cos^2 \theta=1-\sin^2 \theta=1-9/25=16/25$, so $\cos\theta=-4/5$ since $\cos\theta<0$ in Quadrant II. Therefore

$\sin 2\theta=2\sin\theta \cos\theta=2(3/5)(-4/5)=-24/25$.

Pr A Simplify the expression $\cos^4\theta-\sin^4\theta$.

Pr 1 Find $\cos 2\theta$ if $\cos\theta=4/7$.

Pr 2 Find $\sin 2\theta$ if $\sin\theta=5/13$ and $\tan\theta<0$.

Pr 3 Use a half-angle formula to find $\cos\frac{\pi}{8}$.

Pr 4 Simplify the expression $(\sin\frac{\theta}{2}+\cos\frac{\theta}{2})^2.$

Pr 5 Use half-angle formulas to rewrite $\sin^2 \theta \cos^2 \theta$ without using powers of trig functions.

Pr 6 Use half-angle formulas to rewrite $\cos^4 \theta$ without using powers of trig functions.

Pr 7 Find $\sin 4\theta$ if $\sin\theta=4/5$ and $\pi/2<\theta<\pi$.

Pr 8 Find $\cos\frac{\theta}{2}$ if $\sin\theta=-5/8$ and $270^\circ<\theta<360^\circ$.

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