by Dr. Carol JVF Burns (website creator)
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Up to now, we've been thinking of the trigonometric function inputs as angles with units of degrees.

For triangle-based trigonometry (like finding heights of mountains and navigation), this is natural and appropriate.

However, the trigonometric functions are also fundamental in situations where there's not a triangle in sight!
In these cases, a different approach is taken.

For example:

In non-triangle applications, it usually isn't natural to view the trigonometric input as an angle—and there can be disadvantages of doing so.
For example, calculus formulas become more complicated when unit-conversion factors are needed to convert degree input
to a (unit-free) real number.

When using the trigonometric functions, we often want the input to be a plain 'ole real number— just what we're used to with familiar functions
like $\,f(x) = x^2\,.$
Radian measure to the rescue!

You might be thinking—both approaches to trigonometry rely heavily on angles:

How can we get angles ‘out of the picture’ and think of the input as just a real number?

The key to success is to establish a relationship between real numbers (points on a real number line) and the unit circle, as follows:

Associating Real Numbers With Points on the Unit Circle

Think of the real number line as an infinite string:
  • black dot at $\,0\,$ (as shown at right)
  • positive numbers colored green (extending infinitely to the right)
  • negative numbers colored red (extending infinitely to the left)

Pick up this piece of string and wrap it around the unit circle as follows:
  • put the black dot at the point $(1,0)$ on the unit circle
  • wrap the positive numbers counterclockwise (start up)
  • wrap the negative numbers clockwise (start down)
Of course, the green and red will eventually overlap,
as they continue to wrap around and around the circle.
(Only a short piece of green and red ‘string’ are shown here!)

In this way, every real number ends up at (is associated with) a unique point on the unit circle—the so-called ‘terminal point’.
Then, the usual unit circle definitions apply:

As you wrap the real number line ‘string’ around the unit circle,
an angle naturally appears (shown at right in blue).

The real number $\,x\,$ is the radian measure of this angle!

So, what is the radian measure of an angle?
It's the length of the arc that subtends the angle on the unit circle!
DEFINITION radian measure for angles
Let $\,\theta\,$ be any angle, laid off in the standard way on the unit circle:
  • Start at the positive $x$-axis.
  • Positive angles are swept out counterclockwise (start by going up).
  • Negative angles are swept out clockwise (start by going down).
The (appropriately-signed) length of the arc subtended by the angle on the unit circle
is the radian measure of the angle $\,\theta\,.$

Since the circumference of the unit circle is $\,2\pi r = 2\pi(1) = 2\pi\,,$
half the circumference is $\,\pi\,.$

Thus: $$\cssId{s53}{\pi \text{ radians} = 180^\circ}$$
radian measure of $\,\theta\,$
is $\,1\,$
radian measure of $\,\theta\,$
is $\,-2\,$
radian measure of $\,\theta\,$
is $\,\pi\,$
radian measure of $\,\theta\,$
is $\,-\frac{\pi}{2}\,$

Notes on definition:

Converting Between Radian and Degree Measure for Angles

Since $\,\pi\text{ rad} = 180^\circ\,,$ we have: $$ \cssId{s76}{1 = \frac{180^\circ}{\pi \text{ rad}} = \frac{\pi \text{ rad}}{180^\circ}} $$

Conversion between radian and degree measure is then just multiplication by $\,1\,$ in an appropriate form:

To convert from radians to degrees, multiply by $\,\displaystyle\frac{180^\circ}{\pi}\,$: $$ \cssId{s80}{x\text { rad}} \cssId{s81}{= x\text { rad}\cdot\frac{180^\circ}{\pi \text{ rad}}} \cssId{s82}{= \left(\frac{180x}{\pi}\right)^\circ} $$

To convert from degrees to radians, multiply by $\,\displaystyle\frac{\pi}{180^\circ}\,$: $$ \cssId{s85}{x^\circ} \cssId{s86}{= x^\circ\cdot\frac{\pi \text{ rad}}{180^\circ}} \cssId{s87}{= \frac{\pi x}{180} \text{ rad}} $$

Common Radian/Degree Measures You Must Know

There are a handful of radian/degree equalities that you should have on your fingertips, because they're so common:

$180^\circ = \pi \text{ rad}$ $\displaystyle 90^\circ = \frac{\pi}2 \text{ rad}$ $\displaystyle 45^\circ = \frac{\pi}4 \text{ rad}$
$360^\circ = 2\pi \text{ rad}$ $\displaystyle 60^\circ = \frac{\pi}3 \text{ rad}$ $\displaystyle 30^\circ = \frac{\pi}6 \text{ rad}$

Why the name ‘Radian Measure’?

You might think that ‘radian’ is a pretty random name ...
( ... perhaps not any more random than ‘degree’ meaning $\,\frac{1}{360}\,$ of a revolution, or ‘grad’ meaning $\,\frac{1}{400}\,$ of a revolution!)

However, the name ‘radian’ is actually instructive, suggestive, and meaningful, as follows:

Start with an angle $\,\theta\,$ in a circle of radius $\,r\,.$
Suppose the angle subtends an arc of length $\,s\,,$ as shown at right.

We want to find the radian measure of $\,\theta\,.$
To this end, we must scale the circle to the unit circle.
To do this, divide all the lengths through by $\,r\,,$ as shown at right.

Since the radian measure of an angle is the length of the arc it subtends in the unit circle,
we have:
$$ \cssId{s107}{\text{the radian measure of }\theta = \frac{s}{r}} $$
Now, think about the interpretation of the division problem $\displaystyle\,\frac{s}{r}\,.$

The division problem $\displaystyle\,\frac{10}{2}\,$ answers the question ‘How many $2$'s are there in $\,10\,$?’
Answer: $5$

Similarly, the division problem $\displaystyle\,\frac{s}{r}\,$ answers the question ‘How many $r$'s are there in $\,s\,$?’
Answer: the radian measure of the angle!

The radian measure of an angle gives the number of radiuses needed to subtend the angle!

the number of
RADiuses IN
an angle
This knowledge makes it easy to visualize angles that are given in radian measure.

For example, suppose you want to visualize an angle with measure $\,2.5\,$ radians.

Draw a circle of any size, as shown at right.
Imagine cutting a string the length of the radius of the circle.
Lay this string around the circle $\,2.5\,$ times:   $\,1r + 1r + 0.5r = 2.5r\,$

Voila! You just laid off an angle with measure $\,2.5\,$ radians!
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
the Trigonometric Functions
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
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(MAX is 25; there are 25 different problem types.)