# 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."

### Need Graph Paper?

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.

We now choose the portion of this graph from x = 0 to x = π.

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

We now reflect every point on this portion of the cos\ x curve through the line y = x.

The result is the graph y = arccos\ x:

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.)

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

## 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.

As we did previously , if we reflect the indicated portion of y = sin x 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.

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

## 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 this portion of the graph 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.

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:

1. arcsin 0.6294 = sin-1(0.6294) = 0.6808
2. arcsin (-0.1568) = sin-1(-0.1568) = -0.1574
3. arccos (-0.8026) = cos-1(-0.8026) = 2.5024
4. 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:

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:

The graph of y = arccsc x:

The graph extends in the negative and positive x-directions.

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 -10 and 10 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:

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: Which is the correct graph of arccot x?.

### Numerical Exercises

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

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

3. Evaluate sin^-1 0

4. Evaluate sec^-1 2

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

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