is $\sin(\sin^{-1}(x))$ always equal to $x$?

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since $\sin^{-1}(x)$ is the opposite of $\sin(x)$ I think this should be true. however when I input some test numbers into the calculators, sometimes it says error. because of this, I would like to know whether this is true or not.

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2 Answers

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Hint: Consider the numbers which give you error. Then think about what inputs $\sin^{-1}$ accepts. Once this is cleared up, then yes, you will always get $x$ back.

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$y=\sin^{-1}x$ will be defined if $-1\le x\le1$

and $-\dfrac\pi2\le y\le\dfrac\pi2$ using Principal values

then $\sin(y)=x$

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