Question

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

Answer 1

Y=arcsin(sin(π/4))
The exact value of sin (π/4) sin is √2/2
y=arcsin(√2/2)
The exact value of arcsin(√2/ 2) y=π/4y=π/4

Interchange the variables.x=π/4

Answer 2

We are given trigonometry expression.

$\sin^{-1}(\sin(\frac{3\pi}{4}))\neq \frac{3\pi}{4}$

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 $-\frac{\pi}{2}\leq \theta \leq \frac{\pi}{2}$

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

As we know $\sin(\frac{\pi}{4})=\frac{1}{\sqrt{2}}$

Thus, The principle value are not same.

Hence, $\sin^{-1}(\sin(\frac{3\pi}{4}))\neq \frac{3\pi}{4}$