Math Core

Rotate a ray $60^\circ$ , or rotate it $420^\circ$ , and it ends up pointing in exactly the same direction. Angles like these share a terminal side, and trig functions can't tell them apart. In this lesson you'll learn to trade any angle for a simpler one with the same terminal side, and then for a small acute angle that carries almost all of its trig information.

Coterminal angles

A full rotation brings a ray back to where it started. So if you add or subtract one full turn ( $360^\circ$ , or $2\pi$ radians) to an angle, the terminal side doesn't move.

Definition

Coterminal angles

Two angles in standard position are coterminal if they have the same terminal side. Every angle coterminal with $\theta$ can be written as

\theta + 360^\circ k \qquad \text{or} \qquad \theta + 2\pi k,

where $k$ is any integer.

Each angle has infinitely many coterminal angles, one for each integer $k$ . For $60^\circ$ :

\dots,\ -660^\circ,\ -300^\circ,\ 60^\circ,\ 420^\circ,\ 780^\circ,\ \dots

The graph below shows $60^\circ$ and $-300^\circ$ . The first turns counterclockwise a sixth of the way around; the second turns clockwise five sixths of the way. Both stop at the same ray.

60° (solid arc, counterclockwise) and −300° (dashed arc, clockwise) are coterminal.

Finding the coterminal angle in one turn

The most useful coterminal angle is the one between $0^\circ$ and $360^\circ$ (or $0$ and $2\pi$ ). To find it, add or subtract full turns until you land in that interval. For a large angle, divide by $360^\circ$ to see how many whole turns to remove.

Worked example: Coterminal angles in degrees

Find the angle between $0^\circ$ and $360^\circ$ that is coterminal with each angle.

$800^\circ$
$-1000^\circ$

Solutions.

$800 \div 360 \approx 2.2$ , so remove two full turns: $800^\circ - 720^\circ = 80^\circ$ .
The angle is negative, so add full turns. $1000 \div 360 \approx 2.8$ , so three turns are needed to get past zero: $-1000^\circ + 1080^\circ = 80^\circ$ .

Both angles are coterminal with $80^\circ$ (and with each other).

Worked example: Coterminal angles in radians

Find the angle between $0$ and $2\pi$ that is coterminal with $\dfrac{19\pi}{4}$ , and with $-\dfrac{7\pi}{3}$ .

Write $2\pi$ with the same denominator as the angle, then add or subtract.

$2\pi = \dfrac{8\pi}{4}$ . Subtract it twice: $\dfrac{19\pi}{4} - \dfrac{16\pi}{4} = \dfrac{3\pi}{4}$ .
$2\pi = \dfrac{6\pi}{3}$ . Add it twice: $-\dfrac{7\pi}{3} + \dfrac{12\pi}{3} = \dfrac{5\pi}{3}$ .

Reference angles

Once an angle is between $0^\circ$ and $360^\circ$ , you can shrink it one step further. Look at how far its terminal side is from the $x$ -axis.

Definition

Reference angle

The reference angle of an angle $\theta$ in standard position is the acute angle $\theta'$ between the terminal side of $\theta$ and the $x$ -axis. It is always positive and between $0^\circ$ and $90^\circ$ .

The reference angle is measured to the nearest part of the $x$ -axis, never to the $y$ -axis. In the picture below, $150^\circ$ stops $30^\circ$ short of the negative $x$ -axis, so its reference angle is $30^\circ$ .

The reference angle of 150° is the 30° gap between the terminal side and the negative x-axis.

Where the terminal side lands decides how you compute the reference angle.

Reference angle by quadrant

For $\theta$ between $0^\circ$ and $360^\circ$ (or $0$ and $2\pi$ ):

Quadrant of $\theta$	Degrees	Radians
I	$\theta' = \theta$	$\theta' = \theta$
II	$\theta' = 180^\circ - \theta$	$\theta' = \pi - \theta$
III	$\theta' = \theta - 180^\circ$	$\theta' = \theta - \pi$
IV	$\theta' = 360^\circ - \theta$	$\theta' = 2\pi - \theta$

If the angle is outside one turn, first find its coterminal angle in $[0^\circ, 360^\circ)$ .

Quadrantal angles such as $90^\circ$ and $180^\circ$ lie on an axis, so they have no acute gap to measure. You'll handle them separately when you evaluate trig functions.

Worked example: Reference angles

Find the reference angle.

$210^\circ$
$\dfrac{5\pi}{3}$
$-250^\circ$

Solutions.

$210^\circ$ is between $180^\circ$ and $270^\circ$ , so it is in Quadrant III. $\theta' = 210^\circ - 180^\circ = 30^\circ$ .
$\dfrac{5\pi}{3} = 300^\circ$ , so it is in Quadrant IV. $\theta' = 2\pi - \dfrac{5\pi}{3} = \dfrac{\pi}{3}$ .
First find a coterminal angle: $-250^\circ + 360^\circ = 110^\circ$ , which is in Quadrant II. $\theta' = 180^\circ - 110^\circ = 70^\circ$ .

Worked example: Putting both steps together

Find the reference angle of $\dfrac{23\pi}{6}$ .

Step 1: find the coterminal angle. $2\pi = \dfrac{12\pi}{6}$ , and $\dfrac{23\pi}{6} - \dfrac{12\pi}{6} = \dfrac{11\pi}{6}$ .

Step 2: find the quadrant. $\dfrac{11\pi}{6}$ is eleven $30^\circ$ slices, or $330^\circ$ . That's Quadrant IV.

Step 3: measure to the $x$ -axis. $\theta' = 2\pi - \dfrac{11\pi}{6} = \dfrac{\pi}{6}$ .

Common mistake

Don't measure to the $y$ -axis. For $120^\circ$ , the terminal side is $30^\circ$ past the positive $y$ -axis, but the reference angle is $180^\circ - 120^\circ = 60^\circ$ , the gap to the negative $x$ -axis. Also, a reference angle is never negative: for $-45^\circ$ , the reference angle is $45^\circ$ .

Tip

A quick check: a reference angle must be between $0^\circ$ and $90^\circ$ (between $0$ and $\dfrac{\pi}{2}$ ). If your answer is bigger than that, or negative, you used the wrong formula for the quadrant.

Why reference angles matter

Later in this course you'll see that the sine, cosine and tangent of any angle equal those of its reference angle, except possibly for a sign that depends on the quadrant. So $210^\circ$ , $150^\circ$ and $330^\circ$ all share values with $30^\circ$ . Instead of memorizing trig values for every angle, you only need the acute ones, plus the rule for signs.