Answer
$$\theta\approx106.26^\circ$$
Work Step by Step
$$\sin\frac{\theta}{2}=\frac{1}{m}$$
$$m=\frac{5}{4}$$
Replace $m=\frac{5}{4}$ into the formula.
$$\sin\frac{\theta}{2}=\frac{1}{\frac{5}{4}}=\frac{4}{5}$$
- From half-angle identity for sines:
$$\sin\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{2}}$$
Therefore, $$\pm\sqrt{\frac{1-\cos\theta}{2}}=\frac{4}{5}$$
As $\sqrt{\frac{1-\cos\theta}{2}}\ge0$ for all $\theta$, the equation would happen when we pick the positive square root.
$$\sqrt{\frac{1-\cos\theta}{2}}=\frac{4}{5}$$
$$\frac{1-\cos\theta}{2}=\frac{16}{25}$$
$$25(1-\cos\theta)=32$$
$$1-\cos\theta=\frac{32}{25}$$
$$\cos\theta=1-\frac{32}{25}=-\frac{7}{25}$$
which means $$\theta\approx106.26^\circ$$
