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ArbitrLikvidat [17]
3 years ago
10

Explain why sin^-1(sin(3pi/4))does not equal 3pi/4 when y=sin x and y=sin^-1 x are inverses.

Mathematics
1 answer:
Fofino [41]3 years ago
7 0

Because they're not inverses, not exactly. \sin x is not invertible on its entire domain because it's not one-to-one. There are infinitely many values of x such that \sin x=0, for example.

The standard function \sin^{-1}x has a domain of -1\le x\le1 and outputs values between -\dfrac\pi2 and \dfrac\pi2. This means that its inverse, \sin x, is indeed its inverse as long as -\dfrac\pi2\le x\le\dfrac\pi2.

\dfrac{3\pi}4 is larger than \dfrac\pi2 and thus does not fall in the "invertible part" of the domain of \sin x. We have

\sin\dfrac{3\pi}4=\dfrac1{\sqrt2}

which is a value between -1 and 1, so that

\sin^{-1}\left(\sin\dfrac{3\pi}4\right)=\sin^{-1}\left(\dfrac1{\sqrt2}\right)=\dfrac\pi4

If we wanted to recover \dfrac{3\pi}4, we'd have to redefine \sin^{-1}x or define a new inverse function that works on a different branch of the domain of \sin x.

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