Sine, cosine and tangent repeat values, so none of them is one-to-one on its full domain. To reverse them as functions, each is restricted to an interval on which every output occurs at most once.

Principal inverse functions

The inverse sine function

$$\arcsin x$$

returns the angle in $[-\pi/2,\pi/2]$ whose sine is $x$. Its domain is $[-1,1]$.

The inverse cosine

$$\arccos x$$

returns an angle in $[0,\pi]$, also for $x\in[-1,1]$.

The inverse tangent

$$\arctan x$$

returns an angle in $(-\pi/2,\pi/2)$ and is defined for every real $x$.

For example,

$$\arcsin\left(\frac12\right)=\frac{\pi}{6}.$$

Recovering angles

Inverse trigonometric functions turn a known ratio back into an angle. In a right triangle with opposite side $3$ and adjacent side $4$,

$$\tan\theta=\frac34,$$

so the acute angle is

$$\theta=\arctan\left(\frac34\right).$$

Inverse is not reciprocal

The notation $\sin^{-1}x$ is often used for $\arcsin x$, but it does not mean $1/\sin x$. The reciprocal of sine is cosecant.

When solving a trigonometric equation, an inverse function gives a principal angle. Periodicity and symmetry may produce additional solutions outside the principal interval.