6.5 Double-Angle Formulas

Notes

Setting $A = B = \theta$ in the sum formulas gives:

• $\sin 2\theta = 2\sin\theta \cos\theta$.
• $\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$.
• $\displaystyle \tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}$.
• Example: if $\displaystyle \sin\theta = \frac{4}{5}$ and $\theta$ is in Quadrant I, $\displaystyle \cos\theta = \frac{3}{5}$, so $\displaystyle \sin 2\theta = 2 \cdot \frac{4}{5} \cdot \frac{3}{5} = \frac{24}{25}$.