Question

use sum, difference, product, or half-angle formulas to find the exact value of each expression

$\sin \left(\frac{1}{2}{\cos }^{-1}\frac{3}{5}\right)$

Short answer

$\sin \left(\frac{1}{2}{\cos }^{-1}\frac{3}{5}\right)=\frac{1}{\sqrt{5}}$

Step-by-step solution

Step 1.Given expression

$\sin \left(\frac{1}{2}{\cos }^{-1}\frac{3}{5}\right)$

Step 2. Solving 

let

$\frac{1}{2}{\cos }^{-1}\frac{3}{5}=\theta$

So , we need to calculate for

$\sin \theta =?$

$$\begin{gathered} \frac{1}{2}{\cos }^{-1}\frac{3}{5}=\theta \\ {\cos }^{-1}\frac{3}{5}=2\theta \\ -1⩽2\theta ⩽1 \end{gathered}$$

Step 3. Solving for sinθ

$$\begin{gathered} {\cos }^{-1}\frac{3}{5}=2\theta \\ \cos 2\theta =\frac{3}{5} \end{gathered}$$

We know :

role="math" localid="1646555157092" $1-2{\sin }^{2}x=\cos 2x$

Using this :

$$\begin{gathered} \cos 2\theta =\frac{3}{5} \\ 1-2{\sin }^2\theta =\frac{3}{5} \\ 1-\frac{3}{5}=2{\sin }^2\theta \\ \frac{2}{5}=2{\sin }^2\theta \\ \sin \theta =\frac{1}{\sqrt{5}} \end{gathered}$$