Search IntMath
Close

450+ Math Lessons written by Math Professors and Teachers

5 Million+ Students Helped Each Year

1200+ Articles Written by Math Educators and Enthusiasts

Simplifying and Teaching Math for Over 23 Years

# 7. The Inverse Trigonometric Functions

by M. Bourne

In the section Trigonometric Functions of Any Angle, we solved questions like

"Find 2 angles whose cosine is 0.7."

This question involved the use of the cos-1 button on our calculators. We found cos-10.7 and then considered the quadrants where cosine was positive. Remember that the number we get when finding the inverse cosine function, cos-1, is an angle.

Now we turn our attention to all the inverse trigonometric functions and their graphs. It is good to have a sense of these graphs so that you know why there are restrictions on the values that we find on our calculators.

## The Inverse Cosine Function (arccos)

[I have mentioned elsewhere why it is better to use arccos than cos^-1 when talking about the inverse cosine function.]

Let's first recall the graph of y = cos\ x (which we met in Graph of y = a cos x) so we can see where the graph of y = arccos\ x comes from.

Graph of y = cos x.

We now choose the portion of this graph from x = 0 to x = π, shown here by the shaded portion:

Graph of y = cos x with shaded portion 0 <= x <= pi.

The graph of the inverse of cosine x is found by reflecting the chosen portion of the graph of cos x through the line y = x.

Graph of y = cos x and the line y=x.

We now reflect every point on this portion of the cos x curve through the line y = x (I've shown just a few typical points being reflected.)

Reflecting points on the curve through the line y=x.

The result is the graph y = arccos x:

The curve y=arccos x.

See an animation of this process here: Inverse Trigonometric Function Graph Animations.

That's it for the graph - it does not extend beyond what you see here. (If it did, there would be multiple values of y for each value of x and then we would no longer have a function.) I've indicated the "start" and "end" points, (-1, pi) and (1,0) with dots.

NOTE 1: The axes labels have also been reflected. That is, now there are ordinary numbers along the x-axis, and multiples of 0.5pi on the y-axis.

NOTE 2: You will also see "arccos" written as ""acos"", especially in computer programming.

The domain (the possible x-values) of arccos x is

-1 ≤ x ≤ 1

The range (of y-values for the graph) for arccos x is

0 ≤ arccos xπ

Continues below

## The Inverse Sine Function (arcsin)

We define the inverse sine function as

y=arcsin\ x for -pi/2<=y<=pi/2

where y is the angle whose sine is x. This means that

x = sin y

### The graph of y = arcsin x

Let's see the graph of y = sin x first and then derive the curve of y = arcsin x.

Graph of y = sin x, with the portion from -pi/2 to pi/2 highlighted.

As we did previously , if we reflect the indicated portion of y = sin x (the part between x=-pi/2 and x=pi/2) through the line y = x, we obtain the graph of y = arcsin x:

Once again, what you see is what you get. The graph does not extend beyond the indicated boundaries of x and y. I've indicated the "start" and "end" points with dots.

The domain (the possible x-values) of arcsin x is

-1 ≤ x ≤ 1

The range (of y-values for the graph) for arcsin x is

-π/2 ≤ arcsin\ x ≤ π/2

See an animation of this process here:

## The Inverse Tangent Function (arctan)

As a reminder, here is the graph of y = tan x, that we met before in Graphs of tan, cot, sec and csc.

Reflecting the shaded portion of the graph (from x=-pi/2 to pi/2) in the line y = x, we obtain the graph of y = arctan x:

This time the graph does extend beyond what you see, in both the negative and positive directions of x, and it doesn't cross the dashed lines (the asymptotes at y=-pi/2 and y=pi/2).

The domain (the possible x-values) of arctan x is

All values of x

The range (of y-values for the graph) for arctan x is

-π/2 < arctan x < π/2

### Numerical Examples of arcsin, arccos and arctan

Using a calculator in radian mode, we obtain the following:

arcsin 0.6294 = sin-1(0.6294) = 0.6808

arcsin (-0.1568) = sin-1(-0.1568) = -0.1574

arccos (-0.8026) = cos-1(-0.8026) = 2.5024

arctan (-1.9268) = tan-1(-1.9268) = -1.0921

Note that the calculator will give the values that are within the defined range for each function.

## The Inverse Secant Function (arcsec)

The graph of y = sec x, that we met before in Graphs of tan, cot, sec and csc:

The graph of y = arcsec x is obtained by reflecting the shaded portion of the above curve in the line y = x:

:

The curve is defined outside of the portion betweem −1 amd 1. I've indicated the "starting" points (-1,pi) and (1,0) with dots.

The domain of "arc"sec\ x is

All values of x, except −1 < x < 1

The range of arcsec x is

0 ≤ arcsec xπ, "arc"sec\ x ≠ π/2

## The Inverse Cosecant Function (arccsc)

The graph of y = csc x, that we met before in Graphs of tan, cot, sec and csc, looks like this:

Notice there are no values of y between −1 and 1.

Now for the graph of y = arccsc x, which we obtain by reflecting the shaded portion of the above curve in the line y = x:

The graph is not defined between −1 and 1, but extends in the negative and positive x-directions from there.

The domain of arccsc x is

All values of x, except −1 < x < 1

The range of arccsc x is

-π/2 ≤ "arc"csc\ x ≤ π/2, arccsc x ≠ 0

## The Inverse Cotangent Function (arccot)

The graph of y = cot x, that we met before in Graphs of tan, cot, sec and csc is as follows:

Taking the highlighted portion as above, and reflecting it in the line y = x, we have the graph of y = arccot x:

The graph extends in the negative and positive x-directions (it doesn't stop at -8 and 8 as shown in the graph).

So the domain of arccot x is:

All values of x

The range of arccot x is

0 < arccot x < π

### Alternate View

Some math textbooks (and some respected math software, e.g. Mathematica) regard the following as the region of y = cot x that should be used:

This would give the following when reflected in the line y = x:

Graph of y="arccot"\ x; alternative view.

Once again, the graph extends in the negative and positive x-directions.

The domain of arccot x would also be:

All values of x

Using this version, the range of arccot x would be:

-π/2 < "arc"cot\ x ≤ π/2 (arccot x ≠ 0)

See the discussion on this at:

Don't miss the animations of all the graphs on this page here:

### Numerical Exercises

1. Find cos (sin^(-1)0.5)

The value of the portion in brackets is an angle.

Noting the range for inverse sine function, we get

sin^(-1)0.5=pi/6

Hence

cos(sin^(-1)0.5)=cos(pi/6)=0.8660

This final answer is a number, not an angle.

2. Write down the meaning (in words) of y = 2\ sin^-1\ x

y is twice the angle whose sine is x.

3. Evaluate sin^-1 0

sin^-1\ 0 = 0

4. Evaluate sec^-1 2

sec^(-1)2 =pi/3

5. Evaluate cos [tan^-1(-1)]

tan^(-1)(-1)=-pi/4
cos(-pi/4)=1/2sqrt(2) 