Chapter 5, Problem 32RE

Solution

To calculate: General solution of the equation $2{\sin }^{-1}x=\sqrt{2}$ .

The solution is $x\approx 0.6496$ .

Given information:

The equation is $2{\sin }^{-1}x=\sqrt{2}$ .

Calculation:

Divide both sides of the given equation by 2 and simplify.

  $\begin{array}{c}\frac{2{\sin }^{-1}x}{2}=\frac{\sqrt{2}}{2}\\ {\sin }^{-1}x=\frac{1}{\sqrt{2}}\end{array}$

Take sine of both sides of the obtained equation.

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

Use the property $\sin \left({\sin }^{-1}x\right)=x$ and simplify.

  $x=\sin \left(\frac{1}{\sqrt{2}}\right)$

Consequently, there is only one solution for the given equation, which is $x=\sin \left(\frac{1}{\sqrt{2}}\right)$ .

Therefore, the exact answer will be $x\approx 0.6496$ .