Question

Write $\sin \left(2{\cos }^{-1}x\right)$as an algebraic function.

Short answer

The given expression as an algebraic function is$2x\left(\sqrt{1-x^2}\right).$

Step-by-step solution

Step 1. Given Information.

The given trigonometric expression is$\sin \left(2{\cos }^{-1}x\right).$

Step 2. Write the given expression as an algebraic function.

To write the given expression as an algebraic function, let $\theta ={\cos }^{-1}x.$

As we know, $\cos =\frac{b}{h},$here role="math" $b=x\mathrm{and}h=1.$

By using Pythagoras theorem,

$$\begin{gathered} h^2=b^2+p^2 \\ {1}^2=x^2+p^2 \\ \sqrt{1-x^2}=p \end{gathered}$$

Now,

$$\begin{gathered} \sin \left(2\theta \right) \\ =2\sin \theta \cos \theta \\ =2\left(\frac{\sqrt{1-x^2}}{1}\right)\left(\frac{x}{1}\right) \\ =2x\left(\sqrt{1-x^2}\right) \end{gathered}$$