Question

y = sin^-1 x means _____ , where $-1\le x\le 1and-\frac{\mathrm{\pi }}{2}\le y\le \frac{\mathrm{\pi }}{2}$

Short answer

$y={\sin }^{-1}xmeans\mathit{x}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathbf{\left(}\mathit{y}\mathbf{\right)},where-1\le x\le 1and-\frac{\mathrm{\pi }}{2}\le y\le \frac{\mathrm{\pi }}{2}$

Step-by-step solution

Step 1. Given Information:

Given expression: $y={\sin }^{-1}xmeans\mathit{\_}\mathit{\_}\mathit{\_}\mathit{\_}\mathit{\_}\mathbf{\hspace{0.17em}},where-1\le x\le 1and-\frac{\mathrm{\pi }}{2}\le y\le \frac{\mathrm{\pi }}{2}$

We want to fill the gap in given expression.

Step 2. Solution:

We know that

The sin^-1 function is the inverse of the sin function so they cancel each other under predefined conditions.

Applying the sin function on both sides of the given expression we get:

$\sin \left(y\right)=\sin \left({\sin }^{-1}x\right)$

Therefore we get:

role="math" localid="1647506080186" $\sin \left(y\right)=x\mathbf{\hspace{0.17em}},where-1\le x\le 1and-\frac{\mathrm{\pi }}{2}\le y\le \frac{\mathrm{\pi }}{2}$