Half-Angle Identites

L3CALC SCH_CALC

---

Half-angle identities

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

These formulas are derived from the cosine double angle formula.

Sine Half-Angle:

$$\cos 2x=1-2{\sin }^{2}x$$ $$2{\sin }^{2}x=1-\cos 2x$$ $${\sin }^{2}x=\frac{1-\cos 2x}{2}$$ $$\sin x=\pm \sqrt{\frac{1-\cos 2x}{2}}$$

Let $x=\frac{\theta }{2}$, so $2x=\theta$:

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

Cosine Half-Angle

$$\cos 2x=2{\cos }^{2}x-1$$ $$2{\cos }^{2}x=1+\cos 2x$$ $${\cos }^{2}x=\frac{1+\cos 2x}{2}$$ $$\cos x=\pm \sqrt{\frac{1+\cos 2x}{2}}$$

Let $x=\frac{\theta }{2}$:

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

Tangent Half-Angle

$$\tan \frac{\theta }{2}=\frac{\sin \frac{\theta }{2}}{\cos \frac{\theta }{2}}$$

Multiply top and bottom by $2\sin \frac{\theta }{2}$:

$$\tan \frac{\theta }{2}=\frac{2{\sin }^2\frac{\theta }{2}}{2\sin \frac{\theta }{2}\cos \frac{\theta }{2}}$$

From the sine working above, $2{\sin }^2\frac{\theta }{2}=1-\cos \theta$. From the sine double angle formula, $2\sin \frac{\theta }{2}\cos \frac{\theta }{2}=\sin \theta$.

$$\tan \frac{\theta }{2}=\frac{1-\cos \theta }{\sin \theta }$$