Question

Explain why arcsin(sin(3π/4)) ≠ 3π/4 when y = sin(x) and y = arcsin(x) are inverses.

Answer 1

The arcsin function is the inverse of the sin function. However, the arcsin function only returns values in the range [-π/2, π/2]. Therefore, arcsin(sin(3π/4)) is equal to π/4, not 3π/4.

Step-by-step explanation:

The arcsin function is the inverse of the sin function. In other words, if y = sin(x), then x = arcsin(y). However, it is important to note that the arcsin function only returns values in the range [-π/2, π/2].

When evaluating arcsin(sin(3π/4)), we need to find an angle whose sine is equal to sin(3π/4). Since sin(3π/4) = √2/2, we need to find an angle whose sine is √2/2.

Although 3π/4 is an angle whose sine is √2/2, it is not the only angle. In fact, sin(3π/4) = sin(π/4), sin(5π/4), sin(7π/4), etc. The arcsin function will only return the principal value, which is the angle in the range [-π/2, π/2]. Therefore, arcsin(sin(3π/4)) is equal to π/4, not 3π/4.

Answer 2

Short answer:
`\sin x` and
`\arcsin x` are *not* inverses. At least not perfect ones.


`\sin x` is a multi-valued function, i.e. not one-to-one, so it's not invertible.

On the other hand, you can restrict the domain of
`\sin x` to a subset of the standard domain (all real numbers) over which
`\sin x` is not multi-valued. The standard choice for this is to consider
`-\frac\pi2\le x\le\frac\pi2`. If you have a calculator with the
`\arcsin` function built into it, this is the definition it uses. Notice that
`x=\frac{3\pi}4` is outside this restricted domain.

This is why


`\arcsin\left(\sin\frac{3\pi}4\right)=\arcsin\left(\frac1{\sqrt2}\right)=\frac\pi4\\eq\frac{3\pi}4`