Introduction

All the angles we have constructed so far have been inside the range (-360^\circ, 360^\circ). However, we can also construct angles outside this range.

The way we do this involves coterminal angles. Coterminal angles are different angles that appear as the same angle in the coordinate plane. For example, the angles 0^\circ and 360^\circ are coterminal because a rotation of 360^\circ is the same as the rotation of 0^\circ. In general, whenever we add or subtract a multiple of 360^\circ, we get a coterminal angle.

The standard way in which we draw a positive angle \theta in the coordinate plane is as follows:

  • Subtract integer multiples of 360^\circ until we get a coterminal angle that lies in the range [0^\circ, 360^\circ ).

  • Draw the resulting angle in the coordinate plane.

For example, to represent the angle 405^\circ in the coordinate plane, we first subtract 360^\circ\mathbin{:}

\begin{align} 405^\circ - 360^\circ = 45^\circ\quad{\color{green}{\checkmark}} \end{align}

The result, 45^\circ, is a coterminal angle that lies in the range [0^\circ, 360^\circ). So, to draw the angle 405^\circ, we draw the coterminal angle 45^\circ in the coordinate plane.

Note: For every multiple of \color{blue}360^\circ we subtract, we make one complete rotation counter-clockwise around the origin. A rotation of {\color{blue}360^\circ} is the same as the rotation of {\color{blue}0^\circ}, so it doesn't contribute to our final angle.

Here, we subtracted {\color{blue}360^\circ} a single time, so we make a single rotation.

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Example: Finding the Correct Quadrant for Angles that Contain More Than One Full Rotation

The angle \theta = \dfrac {8\pi}3 is represented in the coordinate plane in the usual way. In which quadrant does it lie?

EXPLANATION

First, we find an angle that's coterminal with \dfrac {8\pi}3 by subtracting integer multiples of 2\pi until we get a coterminal angle that lies in the range [0, 2\pi).

\begin{align} \dfrac {8\pi}3 - 2\pi = \dfrac{2\pi}{3}\quad{\color{green}{\checkmark}} \end{align}

So the angle \dfrac {2\pi}3 is coterminal with \dfrac {8\pi}3. Let's draw this coterminal angle in the coordinate plane.

We see that \dfrac{2\pi}{3} lies in quadrant II. Therefore, \dfrac{8\pi}{3} also lies in this quadrant.

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Practice: Finding the Correct Quadrant for Angles that Contain More Than One Full Rotation

The angle $\theta = \dfrac {13\pi}6$ is represented in the coordinate plane in the usual way. In which quadrant does it lie?

a
 II
b
 III
c
 Between quadrants I and II
d
 I
e
 IV
Practice: Finding the Correct Quadrant for Angles that Contain More Than One Full Rotation

The angle $\theta = 400^\circ$ is represented in the coordinate plane in the usual way. In which quadrant does it lie?

a
 I
b
 IV
c
 III
d
 II
e
 Between quadrants I and II
Constructing Negative Angles that Contain More Than One Full Rotation

The method for constructing a negative angle \theta outside the range ( -360^\circ, 360^\circ ) is similar to the method for positive angles, but with one main difference: we add multiples of 360^\circ, instead of subtracting. So the steps are:

  • Add integer multiples of 360^\circ until we get a coterminal angle that lies in the range [0^\circ, 360^\circ ).

  • Draw the resulting angle in the coordinate plane.

For example, we can represent the angle -660^\circ as follows:

\begin{align} -660^\circ + 360^\circ &= -300^\circ \\ -300^\circ + 360^\circ &= 60^\circ\quad{\color{green}{\checkmark}} \end{align}

Now, we draw the angle 60^\circ in the coordinate plane.

Again, for every multiple of 360^\circ that we add, we make one complete rotation around the origin clockwise. A rotation of {\color{blue}360^\circ} is the same as the rotation of {\color{blue}0^\circ}, so it doesn't contribute to our final angle.

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Example: Finding the Correct Quadrant for Negative Angles that Contain More Than One Full Rotation

The angle \theta = -435^\circ is represented in the coordinate plane in the usual way. In which quadrant does it lie?

EXPLANATION

First, we find an angle that's coterminal with -435^\circ by adding integer multiples of 360^\circ until we get a coterminal angle that lies in the range [0^\circ, 360^\circ).

\begin{align} -435^\circ + 360^\circ &= -75^\circ \\[5pt] -75^\circ + 360^\circ &= 285^\circ\quad{\color{green}{\checkmark}} \end{align}

So the angle 285^\circ is coterminal with -435^\circ. Let's draw this coterminal angle in the coordinate plane.

We see that 285^\circ lies in quadrant IV. Therefore, -435^\circ also lies in this quadrant.

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Practice: Finding the Correct Quadrant for Negative Angles that Contain More Than One Full Rotation

The angle $\theta = - \dfrac {9\pi}4$ is represented in the coordinate plane in the usual way. In which quadrant does it lie?

a
 III
b
 I
c
 IV
d
 Between quadrants II and III
e
 II
Practice: Finding the Correct Quadrant for Negative Angles that Contain More Than One Full Rotation

The angle $\theta = -450^\circ$ is represented in the coordinate plane in the usual way. In which quadrant does it lie?

a
 I
b
 II
c
 IV
d
 Between quadrants III and IV
e
 III
Finding Multiple Coterminal Angles

To find a coterminal angle of a given angle \theta, we rotate 360^\circ clockwise or counter-clockwise from \theta. We could do this again and again, each time looping by 360^\circ to find an infinite number of coterminal angles: \theta \pm 360^\circ n,\quad \text{where } n \text{ is any integer.}

For instance, the angles -240^\circ, 120^\circ, 480^\circ, and 840^\circ are all coterminal.

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Example: Finding Coterminal Angles in Degrees

Which angles are coterminal with 240^\circ?

600^\circ, \quad 300^\circ, \quad -480^\circ.

EXPLANATION

To find the coterminal angles of a given angle \theta, we add and subtract integer multiples of 360^\circ\mathbin{.}

Let's start by adding multiples of 360^\circ to 240^\circ\mathbin{:}

\begin{align} 240^\circ + 360^\circ = 600^\circ\quad{\color{green}{\checkmark}} \end{align}

There are no answer choices larger than 600^\circ, so we can stop here.

Now, let's subtract multiples of 360^\circ from 240^\circ\mathbin{:}

\begin{align} 240^\circ - 360^\circ &= -120^\circ \\ 240^\circ - 2\cdot (360^\circ) &= -480^\circ\quad{\color{green}{\checkmark}} \end{align}

There are no answer choices smaller than -480^\circ, so we can stop here.

So, of the given angles, the coterminal angles are 600^\circ and -480^\circ.

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Practice: Finding Coterminal Angles in Degrees

Which of the following angles are coterminal with $\theta = -225^\circ?$

  1. $135^\circ$
  2. $315^\circ$
  3. $-585^\circ$
a
 III only
b
 I and III only
c
 I only
d
 I II, and III
e
 I and II only
Practice: Finding Coterminal Angles in Degrees

Which of the following angles are coterminal with $\theta = 10^\circ?$

  1. $370^\circ$
  2. $-710^\circ$
  3. $550^\circ$
a
 II and III only
b
 II only
c
 I only
d
 I and II only
e
 I, II and III
Example: Finding Coterminal Angles in Radians

Which angle is coterminal with \dfrac {3 \pi} {5}?

\dfrac{4\pi}{5}, \quad -\dfrac{7\pi}{5}, \quad -\dfrac{11\pi}{5}.

EXPLANATION

To find the coterminal angles of a given angle \theta (in radians), we add and subtract integer multiples of 2\pi.

Let's start by adding multiples of 2\pi to \dfrac {3 \pi} {5}\mathbin{:}

\begin{align} \dfrac {3 \pi} {5} + 2\pi = \dfrac {13 \pi} {5} \end{align}

There are no answer choices larger than \dfrac {13 \pi} {5}, so we can stop here.

Now, let's subtract multiples of 2\pi from \dfrac {3 \pi} {5}\mathbin{:}

\begin{align} \dfrac {3 \pi} {5} - 2\pi &= -\dfrac {7 \pi} {5}\quad{\color{green}{\checkmark}} \\ \dfrac {3 \pi} {5} - 2\cdot (2\pi) &= -\dfrac {17\pi} {5} \end{align}

There are no answer choices smaller than -\dfrac {17 \pi} {5}, so we can stop here.

So, of the given angles, the only coterminal angle is -\dfrac{7\pi}{5}.

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Practice: Finding Coterminal Angles in Radians

Which of the following angles are coterminal with $\theta = -\dfrac{2\pi}{3}?$

  1. $\dfrac{4\pi}{3}$
  2. $\dfrac{10\pi}{3}$
  3. $-\dfrac{8\pi}{3}$
a
 I and III only
b
 III only
c
 I only
d
 I and II only
e
 I, II and III
Practice: Finding Coterminal Angles in Radians

Which of the following angles are coterminal with $\theta = \dfrac{7\pi}{6}?$

  1. $\dfrac{19\pi}{6}$
  2. $\dfrac{31\pi}{6}$
  3. $-\dfrac{17\pi}{6}$
a
 I and III only
b
 II only
c
 I only
d
 I, II, and III
e
 I and II only
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