Question

simplify the expression by using a double - angle formula or a half - angle formula.
(a) $sqrt{\frac{1-cos(80^{circ})}{2}}$
(b) $sqrt{\frac{1-cos(4\theta)}{2}}$

Explanation:

Step1: Recall half - angle formula

The half - angle formula for sine is $\sin\frac{\alpha}{2}=\pm\sqrt{\frac{1 - \cos\alpha}{2}}$.

Step2: Simplify part (a)

For $\sqrt{\frac{1-\cos(80^{\circ})}{2}}$, let $\alpha = 80^{\circ}$, then $\sqrt{\frac{1-\cos(80^{\circ})}{2}}=\sin(40^{\circ})$ since $\frac{80^{\circ}}{2}=40^{\circ}$ and we take the positive value as the principal square root in the standard context of trigonometric identities.

Step3: Simplify part (b)

For $\sqrt{\frac{1-\cos(4\theta)}{2}}$, let $\alpha = 4\theta$, then $\sqrt{\frac{1-\cos(4\theta)}{2}}=\sin(2\theta)$ since $\frac{4\theta}{2}=2\theta$.

Answer:

(a) $\sin(40^{\circ})$
(b) $\sin(2\theta)$