# 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^{-1}0.7 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

x≤π

## 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*:

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.

Graph of *y* = tan *x*.

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*:

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) = 0.6808arcsin

^{ }(-0.1568) = sin^{-1}(-0.1568) = -0.1574arccos

^{ }(-0.8026) = cos^{-1}(-0.8026) = 2.5024arctan

^{ }(-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 *y* = sec *x*.

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

Graph of `y="arcsec"\ 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

x≤π, `"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:

Graph of *y* = csc *x*.

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*:

Graph of `y="arccsc"\ 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:

Graph of *y* = 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 `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:

Graph of *y* = cot *x*.

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)`

Answer

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`

Answer

*y* is twice the angle whose sine is *x*.

3. Evaluate `sin^-1 0`

Answer

`sin^-1\ 0 = 0`

4. Evaluate `sec^-1 2`

Answer

`sec^(-1)2 =pi/3`

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

Answer

`tan^(-1)(-1)=-pi/4`

`cos(-pi/4)=1/2sqrt(2)`

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