Inverse Trigonometric Function: Arctangent

Before studying this section, you are encouraged to read Trying to ‘Undo’ Trigonometric Functions.

This section is a copy of Inverse Trigonometric Function: Arcsine, with appropriate changes.
If you've mastered the arcsine section, then this one should be quick and easy!

For a function to have an inverse, each output must have exactly one corresponding input.
Thus, only one-to-one functions have inverses.
The tangent function doesn't have a true inverse, because the tangent function is not one-to-one.

So, to try and define an ‘inverse tangent function’, we do the best we can.
We throw away most of the tangent curve, leaving us with a piece that has three properties:
  • the piece is one-to-one (and hence has an inverse)
  • the piece covers all the outputs from the tangent function (the interval $\,(-\infty,\infty)\,$)
  • the piece is close to the origin
The green piece shown at right satisfies all three of these properties.
This green piece is the restriction of the tangent curve to the interval $\,(-\frac{\pi}{2},\frac{\pi}{2})\,$.

The function that the mathematical community calls ‘the inverse tangent function’
is not actually the inverse of the tangent function, because

the tangent function doesn't have a true inverse.

Instead, the ‘inverse tangent function’ is the inverse of this green piece of the tangent curve.
Several Cycles of
the Graph of the Tangent Function



The tangent function isn't one-to-one;
it doesn't pass a horizontal line test.
So, it doesn't have a true inverse.

To define an ‘inverse tangent function’,
we do the best we can.
Throw away most of the curve—
leave only the green part.

This green part is one-to-one.
This green part does have an inverse.

The inverse of this green part is
what the mathematical community calls
‘the inverse tangent function’.

The arctangent function (precise definition below) is the best we can do in trying to get an inverse of the tangent function.
The arctangent function is actually the inverse of the green piece shown above!

Here's a ‘function box’ view of what's going on:

The tangent function takes a real number (excluding $\,\frac{\pi}{2} + k\pi\,$ for integers $\,k\,$) as an input.
It gives a real number output.

For example (as below),
the output $\,0.5\,$
might come from the tangent function.

When we try to use the tangent function box ‘backwards’, we run into trouble.

The output $\,0.5\,$ could have come
from any of the inputs shown.

However, when we use
the green piece of the tangent curve,
the problem is solved!

Now, there's only one input that works.
(It's the value of the green $\,\color{green}{x}\,$.)

Observe that $\,\color{green}{x}\,$ is in the interval $\,(-\frac{\pi}2,\frac{\pi}2)\,$.


It's a bit of a misnomer, but the arctangent function (precise definition below) is often referred to as the ‘inverse tangent function’.
A better name would be something like ‘the inverse of an appropriately-restricted tangent function’.
(It's no surprise, however, that people don't say something that long and cumbersome.)

So, what exactly is $\,\arctan 0.5\,$?

$\,\arctan 0.5\,$ is the number in the interval $\,(-\frac{\pi}{2},\frac{\pi}{2})\,$ whose tangent is $\,0.5\,$

What exactly IS $\,\arctan x\,$?

More generally, let $\,x\,$ be any real number.
Then:

$\,\arctan x\,$ is the number in the interval $\,(-\frac{\pi}{2},\frac{\pi}{2})\,$ whose tangent is $\,x\,$

In my own mind (author Dr. Carol Burns speaking here), the words I say are:

$\,\arctan x\,$ is the number between $\,-\frac{\pi}{2}\,$ and $\,\frac{\pi}{2}\,$ whose tangent is $\,x\,$
I personally know the endpoints are not included, so this doesn't confuse me.
However, the word ‘between’ is ambiguous—it can include the endpoints or not, depending on context.
It can be clarified by saying:
$\,\arctan x\,$ is the number between $\,-\frac{\pi}{2}\,$ and $\,\frac{\pi}{2}\,$ (not including the endpoints)
whose tangent is $\,x\,$
... but then it loses its simplicity. Ah—issues with language. Choose words that work for you!

Precise Definition of the Arctangent Function

The precise definition of the arctangent function follows.
It can look a bit intimidating—the notes following the definition should help.

DEFINITION the arctangent function, denoted by  $\,\arctan\,$  or  $\,\tan^{-1}\,$
Let $\,x\,$ be a real number.

Using the notation ‘$\,\arctan\,$’ for the arctangent function: $$ y = \arctan x\ \ \ \ \ \text{if and only if}\ \ \ \ \bigl(\ \tan y = x\ \ \text{AND}\ \ -\frac{\pi}{2} \lt y\lt \frac{\pi}{2}\ \bigr) $$
Using the notation ‘$\,\tan^{-1}\,$’ for the arctangent function: $$ y = \tan^{-1} x\ \ \ \ \ \text{if and only if}\ \ \ \ \bigl(\ \tan y = x\ \ \text{AND}\ \ -\frac{\pi}{2} \lt y\lt \frac{\pi}{2}\ \bigr) $$

Notes on the Definition of the Arctangent Function:

Example: Find the Exact Value of $\,\arctan(-1/\sqrt 3)\,$
(use both the unit circle and a special triangle)

  • Using the degree definition:
    $\,\arctan (-1/\sqrt 3)\,$ is the angle between $\,-90^\circ\,$ and $\,90^\circ\,$ whose tangent is $\,-\frac{1}{\sqrt 3}\,$.
  • As needed, review information about the size and sign of the tangent function.
  • Draw a unit circle.
    Since we want an angle between $\,-90^\circ\,$ and $\,90^\circ\,$ whose tangent is negative, the angle is in quadrant IV.
    Since we want an angle whose tangent has size $\,\frac{1}{\sqrt 3} \approx 0.58\,$, make the red segment have this length.
    The (negative) angle shown is therefore $\,\arctan(-1/\sqrt 3)\,$.
  • Does any special triangle tell us an acute angle whose tangent is $\,\frac{1}{\sqrt 3}\,$?
    Yes! The tangent of $\,30^\circ\,$ is $\,\frac{1}{\sqrt 3}\,$.
  • Thus, $\,\arctan(-1/\sqrt{3}) = -30^\circ\,$.
  • Using radian measure, $\,\arctan(-1/\sqrt{3}) = -\frac{\pi}6\,$.
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Inverse Trigonometric Function Problems: All Mixed Up


On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40 41 42 43 44  
AVAILABLE MASTERED IN PROGRESS

(MAX is 44; there are 44 different problem types.)
Want textboxes to type in your answers? Check here: