solve the equation for x, where x is restricted to the given interval. y = sin(x + 1), for x in -pi/2 - 1…

Explanation:

Step1: Apply inverse - sine function

Given $y = \sin(x + 1)$. To isolate $x$, we take the inverse - sine of both sides. Since $x\in\left[-\frac{\pi}{2}-1,\frac{\pi}{2}-1\right]$, then $x + 1\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$, and the inverse - sine function $\sin^{-1}$ is well - defined. So, $\sin^{-1}(y)=x + 1$.

Step2: Solve for x

Subtract 1 from both sides of the equation $\sin^{-1}(y)=x + 1$. We get $x=\sin^{-1}(y)-1$.

Answer:

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