3.9 Inverse trigonometric function

Consider the equation: \text{sin }x=\dfrac{1}{\sqrt{2}}

We would like to find the solutions to this equation, shown as the points of intersection on the graph below. To begin with we are going to determine an inverse function for sine, which will help us find one of these solutions. We will then look at how to recover the other solutions in a later lesson.

The graph of an inverse function can be found by reflecting the original function about the line y=x. The graph below shows that reflection applied to the graph of y=\text{sin }x.

The graph of y=\text{sin }x reflected about the line y=x.

Notice, however, that this graph has multiple y-values for a given x-value, so it is not a function.

In order to find an inverse function for \text{sin }x we will need to restrict the domain of the function to a portion that meets the following criteria:

1. It only has one y-value for each x-value (it is one-to-one).
2. It covers the full range of values of \text{sin }x.

There are infinitely many ways we could restrict the domain that would satisfy these criteria, but the domain that is typically chosen by mathematicians to work with is \left[-\dfrac{\pi}{2},\,\dfrac{\pi}{2}\right]. When just this part of the curve is reflected about y=x we obtain the inverse function for \text{sin }x, which we represent as \text{sin}^{-1}(x) or \text{arcsin}(x).

We can now rewrite our original equation using the inverse sine function:

Notice that because we needed to restrict the domain of \text{sin }x to obtain an inverse function, we have only found the one solution between -\dfrac{\pi}{2} and \dfrac{\pi}{2}. We will look at how to use this single solution to find the other solutions to the equation later.

We can find inverse functions for \text{cos }x and \text{tan }x in a similar way; we first restrict their domains using the above criteria, then reflect the remaining graph about the line y=x.

The domain of \text{cos }x is restricted to [0,\,\pi] to produce the inverse function, which we represent as \text{cos}^{-1}(x) or \text{arccos}(x).

The domain of \text{tan }x is restricted to \left(-\dfrac{\pi}{2},\,\dfrac{\pi}{2}\right) to produce the inverse function, which we represent as \text{tan}^{-1}(x) or \text{arctan}(x).

Examples

The inverse trigonometric functions, \text{arcsin},\,\text{arccos} and \text{arctan} are notated on calculators, somewhat confusingly, as \text{sin}^{-1},\,\text{cos}^{-1} and \text{tan}^{-1} respectively.

It is important to realize that the notation refers to the function inverses and not to the multiplicative inverses or reciprocal functions.

Functions and their inverses have the property that, for a function f, it is always true that f(f^{-1}(x))=x and also, f^{-1}(f(x))=x.

In the case of the trigonometric functions, this means

Given a number that is in the range of either the sine, cosine or tangent function, we may ask the question, 'What is the number or angle that has this sine, cosine or tangent'. This is the situation in which we use the inverse trigonometric functions.

For example, the number 0.443\,52 is in the range of the cosine function. That is, there is a number \theta such that \text{cos }\theta = 0.443\,52.

We may wish to find the \theta whose cosine is 0.443\,52 and for this, we apply the inverse cosine function, \text{cos}^{-1}(0.443\,52) and obtain the result \theta=63.67\degree.

Examples