# Chapter 5 - Trigonometric Identities - Section 5.6 Half-Angle Identities - 5.6 Exercises - Page 243: 59

$$\theta\approx106.26^\circ$$

#### 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 $$\theta\approx106.26^\circ$$

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