6.6 Half-Angle Formulas

Notes

Solving the double-angle formula for the half angle gives:

• $\displaystyle \sin\frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos\theta}{2}}$ and $\displaystyle \cos\frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos\theta}{2}}$.
• $\displaystyle \tan\frac{\theta}{2} = \frac{\sin\theta}{1 + \cos\theta} = \frac{1 - \cos\theta}{\sin\theta}$.
• Choose the sign from the quadrant of $\displaystyle \frac{\theta}{2}$, not of $\theta$.
• Example: $\displaystyle \cos 15^\circ = \sqrt{\frac{1 + \cos 30^\circ}{2}} = \sqrt{\frac{2 + \sqrt{3}}{4}} = \frac{\sqrt{2 + \sqrt{3}}}{2}$.