5.4 Double-Angle and Half-Angle Identities

Notes

• $\sin 2\theta = 2\sin\theta\cos\theta$.
• $\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$.
• $\displaystyle \tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}$.
• Half-angle: $\displaystyle \sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}$ and $\displaystyle \cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}}$. Choose the sign from the quadrant of $\displaystyle \frac{\theta}{2}$.
• Example: if $\displaystyle \sin\theta = \frac{3}{5}$ and $\theta$ is in quadrant I, $\displaystyle \sin 2\theta = 2 \cdot \frac{3}{5}\cdot\frac{4}{5} = \frac{24}{25}$.