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:
  • 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 smallest angle beween the $x$-axis and the terminal (ending) side of $\,\theta\,$
is called the reference angle for $\,\color{blue}{\theta}\,$.

a reference angle
for a positive angle

a reference angle
for a negative angle

Properties of the Reference Angle:

Examples of Reference Angles

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:

All twenty examples below have a reference angle of $\,37^\circ\,$. (I had a lot of fun with the last two!)

Reference Angles for Multiples of $\,90^\circ\,$

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 $0^\circ$.

Size and Sign

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 $\,7\,$.

Finding Reference Angles and Quadrants

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:

Therefore, it's useful to have an efficient way to find reference angles and quadrants.

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.

Renaming Trigonometric Values using Reference Angle/Quadrant

Let $\,\theta = 1747^\circ\,$.

  1. Cut $\,\theta\,$ down to size: that is, get an angle between $\,-180^\circ\,$ and $\,180^\circ\,$ with the same terminal point.
  2. Find the reference angle and quadrant for $\,\theta\,$.
  3. Use the reference angle to find the size of $\,\sin \theta\,$ and $\,\cos \theta\,$.
  4. Use the quadrant to find the sign of $\,\sin \theta\,$ and $\,\cos \theta\,$.
  5. Rename $\,\sin 1747^\circ\,$ and $\,\cos 1747^\circ\,$ using size (reference angle) and sign (quadrant).

  1. There are a bunch of ‘extra rotations’ in $\,1747^\circ\,$.   How many?   Well, $\displaystyle\frac{1747^\circ}{360^\circ} \approx 5\,$ (rounded to the nearest whole number).
    Then, $1747^\circ - 5\cdot 360^\circ = -53^\circ$, which is between $\,-180^\circ\,$ and $\,180^\circ\,$.

  2. Clearly, $-53^\circ\,$ lies in quadrant IV;   the reference angle is $\,53^\circ\,$.
  3. The reference angle determines the size of the trigonometric values.
    Since the reference angle for $\,\theta = 1747^\circ\,$ is $\,53^\circ\,$:
    the size of $\,\sin 1747^\circ\,$ is $\,\sin 53^\circ\,$;
    the size of $\,\cos 1747^\circ\,$ is $\,\cos 53^\circ\,$
  4. The quadrant determines the sign of the trigonometric values.
    In quadrant IV, sine ($y$-value) is negative and cosine ($x$-value) is positive
  5. Thus:
    $\,\sin 1747^\circ = -\sin 53^\circ\,$
    $\,\cos 1747^\circ = \cos 53^\circ\,$

Renaming Trigonometric Values using Reference Angle/Quadrant

Rename $\,\sin (-2252^\circ)\,$ and $\,\cos (-2252^\circ)\,$ using reference angle and quadrant.


$\displaystyle\frac{2252^\circ}{360^\circ} \approx 6\,$

$-2252^\circ + 6\cdot 360^\circ = -92^\circ$

quadrant: III (both sine and cosine are negative)

reference angle: $\,180^\circ - 92^\circ = 88^\circ$

$\,\sin (-2252^\circ) = -\sin 88^\circ\,$
$\,\cos (-2252^\circ) = -\cos 88^\circ\,$

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Radian Measure
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19  

(MAX is 19; there are 19 different problem types.)