A possible value of [14^-1√638] is

Question

A possible value of $	an\left[\dfrac14\sin^{-1}\dfrac{\sqrt{63}}{8}ight]$ is

Accepted Answer

Let $\theta=\sin^{-1}\dfrac{\sqrt{63}}{8}$: $\sin\theta=\dfrac{\sqrt{63}}{8},\ \cos\theta=\dfrac18$.

$$\cos\frac{\theta}{2}=\sqrt{\frac{1+\cos\theta}{2}}=\sqrt{\frac{1+\frac18}{2}}=\sqrt{\frac{9}{16}}=\frac34.$$

$$\tan\frac{\theta}{4}=\sqrt{\frac{1-\cos\frac{\theta}{2}}{1+\cos\frac{\theta}{2}}}=\sqrt{\frac{1-\frac34}{1+\frac34}}=\sqrt{\frac17}=\frac{1}{\sqrt7}.$$

Solutions · 1