Question

Explain why sin^-1(sin(3pi/4))=/=3pi/4 when y=sin x and y=sin^-1 x are inverses.

Answer 1

We are given trigonometry expression.

`\sin^(-1)(\sin((3\pi)/(4)))\\eq (3\pi)/(4)`

when inverse of
`y=\sin x` is
`y=\sin^(-1)x`

Inverse function gives principle value of angle.

Here we have sin inverse function whose principle value is
`-(\pi)/(2)\leq \theta \leq (\pi)/(2)`

The value of y must be lie between principle value. This changes because of exact value of sine. If we find
`\sin(3\pi/4)=(1)/(√(2))`

As we know
`\sin((\pi)/(4))=(1)/(√(2))`

Thus, The principle value are not same.

Hence,
`\sin^(-1)(\sin((3\pi)/(4)))\\eq (3\pi)/(4)`