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."

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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 cos x

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

Portion of cos x graph

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

cos x and y=x graphs

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

Reflect cos x in line y=x

The result is the graph `y = arccos\ x`:

Graph of 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.

Portion of sin x graph

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:

Graph of 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.

Graph of tan x

Reflecting this portion of the graph in the line y = x, we obtain the graph of y = arctan x:

Graph of 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 answers in each case are angles (in radians).

The Inverse Secant Function (arcsec)

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

Graph of sec x

The graph of y = arcsec x:

Graph of 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:

Graph of csc x

The graph of y = arccsc x:

Graph of 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:

Graph of cot x

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

Graph of 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:

Graph of cot x

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

Alternate graph of arccot 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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