Chapter 5.5, Problem 115E

To determine

Torepresent:Thetrigonometric expression $\sin \left(2\arcsin x\right)$ as an algebraic expression.

Answer to Problem 115E

The trigonometric expression $\sin \left(2\arcsin x\right)$ can be represented as $2x\cdot \sqrt{1-x^2}$ .

Explanation of Solution

Given information:

The given trigonometric expression is $\sin \left(2\arcsin x\right)$ .

Calculation:

Let angle $\arcsin x=u$ . Solve the trigonometric identities to find the other sides of the triangle.

  $\begin{array}{r}\arcsin x=u\\ \sin u=\frac{x}{1}\end{array}$

From the above, the perpendicular is $x$ and the hypotenuse is 1. So, the adjacent side will be $\sqrt{1-x^2}$ according to the Pythagoras theorem.

So, $\cos u=\frac{\sqrt{1-x^2}}{1}=\sqrt{1-x^2}$ .

  

Solve the trigonometric expression.

  $\begin{array}{c}\sin \left(2\arcsin x\right)=\sin 2u\\ =2\sin u\cos u\left\{\sin 2\theta =2\sin \theta \cos \theta \right\}\end{array}$

Substitute $\sqrt{1-x^2}$ for $\cos u$ and $x$ for $\sin u$ in the above expression.

  $\sin \left(2\arcsin x\right)=2x\cdot \sqrt{1-x^2}$

Therefore, thetrigonometric expression $\sin \left(2\arcsin x\right)$ can be represented as $2x\cdot \sqrt{1-x^2}$ .