Inverses of Restricted Trig Functions

A full sine curve is many-one: different angles give the same output. It has an inverse only after restricting its domain to a one-one part, where each output comes from just one input.

Consider $f(x)=2\sin(x/2)+1$ for $-\pi\le x\le\pi$. The inside angle runs from $-\pi/2$ to $\pi/2$, where sine rises without turning back.

Any horizontal line meets this restricted curve at most once. Its minimum is −1 and maximum is 3. This range becomes the inverse's domain.

Write the output as $y$, then swap input and output:

Subtract 1, divide by 2, and use inverse sine. It returns the correct angle because $y/2$ lies in its principal range.

The inverse has domain $-1\le x\le3$ and range $-\pi\le f^{-1}(x)\le\pi$. Check: $f(0)=1$, and $f^{-1}(1)=0$. Graph points swap coordinates, so the inverse is a reflection in $y=x$.

Suppose $g(x)=\cos x$ is restricted to $\pi\le x\le2\pi$. It is also one-one, but inverse cosine normally returns an angle between 0 and $\pi$. Use the matching angle on the given branch instead. Reflection across the horizontal axis keeps the horizontal coordinate, so cosine stays the same:

Its domain is $-1\le x\le1$.

For example, $g^{-1}(0)=3\pi/2$, not $\pi/2$. Always check the original domain before choosing the inverse expression.