Question

If sin^-1 (x+3) = -pi/6 , x = ?

Answer 1

First recall that
`\sin^(-1)x` is defined only for
`-1\le x\le1`, which means
`\sin^(-1)(x+3)` requires that
`2\le x+3\le4`.

Over this domain,
`\sin^(-1)(x+3)` will be invertible, and so taking the sine of both sides, we have


`\sin^(-1)(x+3)=-\frac\pi6\implies\sin\left(\sin^(-1)(x+3)\right)=\sin\left(-\frac\pi6\right)`

`x+3=-\sin\frac\pi6`

`x+3=-\frac12`

`x=-\frac72`