We are already familiar with certain operations having inverse operations that "undo" them - such as addition and subtraction, or multiplication and division. In a similar manner, we can define inverse trigonometric functions to be functions which reverse, or "undo", the existing trigonometric functions.

The functions sine, cosine, and tangent each take an angle measure as input and return a side ratio as output.

For a given angle \theta we have:

The inverse functions work in reverse, taking in the ratio of two known sides as input and returning a missing angle measure as output.

So the three inverse trigonometric functions are: \sin^{-1}\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)= \theta \qquad \cos^{-1}\left(\frac{\text{adjacent}}{\text{hypotenuse}}\right) = \theta \qquad \tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right) = \theta

We can use these inverse trigonometric functions to determine an unknown angle measure when we know a pair of side lengths of a right-triangle.

As these inverse functions "undo" the original functions, we have\sin^{-1}\left(\sin\left(x\right)\right) = x \qquad \cos^{-1}\left(\cos\left(x\right)\right) = x \qquad \tan^{-1}\left(\tan\left(x\right)\right) = xand\sin\left(\sin^{-1}\left(x\right)\right) = x \qquad \cos\left(\cos^{-1}\left(x\right)\right) = x \qquad \tan\left(\tan^{-1}\left(x\right)\right) = x

It is important to note that while the notation used to represent these inverse trigonometric functions looks like a power of -1, these functions are not the same as the reciprocals of the trigonometric functions. That is:

Worked examples