Up to now, we've been thinking of the trigonometric function inputs as angles
with units of degrees.
For trianglebased 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 nontriangle 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 unitconversion factors are needed
to convert degree input
to a (unitfree) 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:
The key to success is to establish a relationship between real numbers (points on a real number line) and the unit circle, as follows:
Think of the real number line as an infinite string:

Pick up this piece of string and wrap it around the unit circle as follows:
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 socalled ‘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:
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: $$\cssId{s53}{\pi \text{ radians} = 180^\circ}$$ 
is $\,1\,$ 
is $\,2\,$ 
is $\,\pi\,$ 
is $\,\frac{\pi}{2}\,$ 
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:
FROM RADIANS TO DEGREES:
To convert from radians to degrees, multiply by $\,\displaystyle\frac{180^\circ}{\pi}\,$:
$$
\cssId{s80}{x\text { rad}}
\cssId{s81}{= x\text { rad}\cdot\frac{180^\circ}{\pi \text{ rad}}}
\cssId{s82}{= \left(\frac{180x}{\pi}\right)^\circ}
$$
FROM DEGREES TO RADIANS:
To convert from degrees to radians, multiply by $\,\displaystyle\frac{\pi}{180^\circ}\,$:
$$
\cssId{s85}{x^\circ}
\cssId{s86}{= x^\circ\cdot\frac{\pi \text{ rad}}{180^\circ}}
\cssId{s87}{= \frac{\pi x}{180} \text{ rad}}
$$
There are a handful of radian/degree equalities that you should have on your fingertips, because they're so common:
$180^\circ = \pi \text{ rad}$  $\displaystyle 90^\circ = \frac{\pi}2 \text{ rad}$  $\displaystyle 45^\circ = \frac{\pi}4 \text{ rad}$ 
$360^\circ = 2\pi \text{ rad}$  $\displaystyle 60^\circ = \frac{\pi}3 \text{ rad}$  $\displaystyle 30^\circ = \frac{\pi}6 \text{ rad}$ 
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! 
measure: 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! 

On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
