In the previous section,
Special Triangles and Common Trigonometric Values,
we kept saying
‘the (acute) angle between the $x$axis and the hypotenuse
(of the special triangle)’.
Quite a mouthful!
It's a really important angle, and in this section we'll give it a shorter name!
DEFINITION
reference angle for $\,\theta$
Let $\,\theta\,$ (the ‘original’ angle) be any angle, laid off in the standard way:
is called the reference angle for $\,\color{blue}{\theta}\,.$ 
Example: a reference angle for a positive angle 
Example: a reference angle for a negative angle 
There's a great applet for exploring reference angles at
Math Open Reference.
The screenshots below were made from this applet, and have the following features:
Angles that are multiples of $\,90^\circ\,$ have terminal points on the $x$axis or $y$axis.
In particular:
$\theta = 90^\circ$ The reference angle for $\theta$ is $90^\circ$. 
$\theta = 11\cdot90^\circ = 990^\circ$ The reference angle for $\theta$ is $90^\circ$. 
$\theta = 0^\circ$ The reference angle for $\theta$ is $0^\circ$. 
$\theta = 6\cdot 90^\circ = 540^\circ$ The reference angle for $\theta$ is $0^\circ$. 
Every real number has a size.
Size is distance from zero, or absolute value.
Size is nonnegative (greater than or equal to zero).
For example, both $\,5\,$ and $\,5\,$ have size $\,5\,.$
The number $\,0\,$ has size $\,0\,.$
Every real number except zero has a sign (plus or minus).
Numbers to the right of zero are positive.
Numbers to the left of zero are negative.
The number zero has no sign.
Zero is not positive.
Zero is not negative.
Size and sign together uniquely identify every nonzero real number.
What real number has size $\,3\,$ and is negative?
The number $\,3\,.$
What real number has size $\,7\,$ and is positive?
The number $\,7\,.$
If the terminal point is on the $x$axis or $y$axis, then the trigonometric
values are extremely easy to find.
For example, if the terminal
point is $\,(1,0)\,,$ then the cosine is $\,1\,,$ the sine is $\,0\,,$
and the other trigonometric values follow from their definitions.
For any angle that doesn't land on an axis, the following two pieces of information are sufficient to find all the trigonometric values:
DISCLAIMER:
When dealing with angles and their reference angles, I'll often say things like
‘an angle in quadrant II’ instead of the
more precise ‘an angle with terminal point in quadrant II’.
What follows is a way to find reference angles and quadrants.
It certainly isn't the only way, but it works.
It allows you to use smaller (more easily identifiable) numbers than some other methods.
Let $\,\theta = 1747^\circ\,.$
Rename $\,\sin (2252^\circ)\,$ and $\,\cos (2252^\circ)\,$ using reference angle and quadrant.
SOLUTION:
$\displaystyle\frac{2252^\circ}{360^\circ} \approx 6\,$
$2252^\circ + 6\cdot 360^\circ = 92^\circ$
quadrant: III (both sine and cosine are negative)
reference angle: $\,180^\circ  92^\circ = 88^\circ$
$\,\sin (2252^\circ) = \sin 88^\circ\,$
$\,\cos (2252^\circ) = \cos 88^\circ\,$
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
