Answer
$$\theta\approx83.62^\circ$$
Work Step by Step
$$\sin\frac{\theta}{2}=\frac{1}{m}$$
$$m=\frac{3}{2}$$
Replace $m=\frac{3}{2}$ into the formula.
$$\sin\frac{\theta}{2}=\frac{1}{\frac{3}{2}}=\frac{2}{3}$$
- From half-angle identity for sines:
$$\sin\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{2}}$$
Therefore, $$\pm\sqrt{\frac{1-\cos\theta}{2}}=\frac{2}{3}$$
As $\sqrt{\frac{1-\cos\theta}{2}}\ge0$ for all $\theta$, the equation would happen when we pick the positive square root.
$$\sqrt{\frac{1-\cos\theta}{2}}=\frac{2}{3}$$
$$\frac{1-\cos\theta}{2}=\frac{4}{9}$$
$$9(1-\cos\theta)=8$$
$$1-\cos\theta=\frac{8}{9}$$
$$\cos\theta=1-\frac{8}{9}=\frac{1}{9}$$
which means $$\theta\approx83.62^\circ$$
