Coterminal Angles-Explanation and Examples

Coterminal angles are angles that have the same terminal side.

The terminal side is the second ray on an angle. These angles look the same on a plane, but their measure may not be the same because of the measuring direction or because the angle represents more than one circuit. 

Coterminal angles are important in trigonometry and calculus. They also have applications in science, especially engineering.

Take a moment to review angles before you continue reading this article.

What are Coterminal Angles?

Coterminal angles are angles that share a terminal side but may not have the same angle measure. The terminal side is the second ray of an angle measured.  Typically, coterminal angles refer to angles on the unit circle.

Such angles are measured from an initial side, which extends from the middle to the right along the x-axis to the terminal angle, which is another radius of the unit circle. 

Coterminal angles may not have the same measure because of the measuring direction. That is, one angle could be measured clockwise and the other counterclockwise. Additionally, one angle may represent more than one circuit around the unit circle. That is, its measure may exceed $360$ degrees or $2\pi$ radians.

How To Find Coterminal Angles

There are two main ways to find coterminal angles: the first way involves negative angles, and the second way involves adding one or more circuits to the “standard” angle.

In the first method, instead of measuring the angle counterclockwise, measure it clockwise and add a negative sign. In the second method, add one or more circuits to the angle measured from the initial angle counterclockwise to the terminal angle.

Take note that it is possible to combine these two methods.

Negative Coterminal Angles

The first way to find a coterminal angle is to find the negative coterminal angle. This is the angle found by measuring clockwise from the initial side to the terminal side instead of counterclockwise. To note the direction that the angle was measured, include a negative sign.

Take note that the negative coterminal angle and the standard angle together form a full angle. This will measure $360$ degrees or $2\pi$ radians.

Therefore, the formula for the negative coterminal angle is:

$-(360-\alpha)$,

Where $\alpha$ is the standard angle.

In radians, this is:

$-(2\pi-\alpha)$.

Positive Coterminal Angle

A positive coterminal angle is the standard angle plus one or more full circles. Imagine you see a clock with the minute hand on the twelve. Later, you see the same clock with the minute hand on the one.

How much time has passed? Five minutes? An hour and five minutes? Two hours and five minutes? Seventy hours and five minutes?

In any case, five minutes plus some whole number of hours has passed.

Similarly, positive coterminal angles are the standard angle plus $360$ degrees or $2\pi$ radians.

As a formula, this is:

$360n+\alpha$,

Where $\alpha$ is the standard angle measure in degrees and $n$ is an integer.

In radians, this is:

$2n\pi+\alpha$.

Take note that $n$ can be negative. When this happens, the coterminal angle is the negative coterminal angle plus one or more circles.

Negative Coterminal Angles and Trigonometry

Since coterminal angles essentially look the same in the Cartesian plane, trigonometric functions’ values at coterminal angles are the same.

Take note, however, that these are not the only situations under which a trigonometric function may have the same output for two different inputs. Therefore, knowing that the value of a trig function for two angles is not enough information to say that the two angles are coterminal. But knowing two angles are coterminal is enough information to say that the angles will have the same values for any trig function.

Examples

This section covers common examples of problems involving coterminal angles and their step-by-step solutions.

Example 1

Find one negative coterminal angle and one positive coterminal angle for the standard angle $60$ degrees.

Solution

A negative coterminal angle will be one that is measured clockwise, and a positive coterminal angle will be one that is measured more than once around the unit circle.

Using the formulas above, a negative coterminal angle is $-(360-60) = -300$ degrees.

A positive coterminal angle is $360(2)+60 = 720+60 = 780$ degrees.

Take note, however, that there are infinitely possible answers.

Example 2

What is the standard angle equal to $458$ degrees?

Solution

This is a positive angle with a measure greater than that of a full angle, $360$ degrees. Therefore, it represents more than one circuit around the unit circle.

To find its standard angle, subtract $360$ degrees as many times as necessary until the difference is a value greater than or equal to zero and less than $360$.

In this case, one iteration does the trick, which is $458-360 = 98$ degrees. This is between $0$ and $360$, so it is the measure of the standard angle coterminal with $458$ degrees.

Example 3

What is the standard angle equal to $-\frac{5}{4}\pi$ radians?

Solution

This is a negative angle, which means it is measured clockwise from the initial angle to the terminal angle.

To find its corresponding standard angle, find the difference between the absolute value of this angle and a full angle. In radians, such an angle is $2\pi$ radians.

Therefore, $2\pi-\frac{5}{4}\pi = \frac{3}{4}\pi$. This is the measure of the coterminal standard angle.

Example 4

What is the standard angle equal to $-894$ degrees? What is the relationship between the cosine of this angle and the cosine of $-894$?

Solution

This angle is both negative and greater than a full angle. Therefore, in this case, instead of subtracting $360$ degrees, add $360$.

One time, however, is not enough. $-894+360 = -534$ degrees, which is still greater than a full angle. Doing this again, however, results in $-534+360 = -174$ degrees. Since this is between $0$ and $360$, there is no need to add any more.

Now use the formula for finding the standard angle given a negative angle. This is $-(360+\alpha)$. In this case, that is $-(360-174) = 186$ degrees.

The cosine of $186$ and the cosine of $-894$ will be the same because the angles are coterminal. In this case, the cosine of both angles is approximately $-0.99$.

Example 5

Write formulae expressing all possible coterminal angles equal to $45$ degrees in both radians and degrees.

Solution

First, convert the angle to radians. $\frac{45}{360} = \frac{x}{2\pi}$. Solving for $x$ yields $x=\frac{1}{8}$. Therefore, in radians, the angle is $\frac{\pi}{4}$.

Next, consider positive coterminal angles. All of these will be a whole number multiple of $360$ degrees or a whole number multiple of $2\pi$ radians. Therefore, the formulae for positive coterminal angles is $45+360n$, where $n$ is a natural number, and $\frac{\pi}{4}+2n\pi$, where $n$ is again a natural number.

After this, consider the negative coterminal angle. When using degrees, a negative coterminal angle is $-(360-45) = 45-360 = -315$ degrees. In radians, the negative coterminal angle is $2\pi-\frac{\pi}{4} = \frac{7\pi}{4}$.

Finally, consider all of the negative coterminal angles. That is, consider the negative coterminal angles that represent more than one circuit. In degrees, these will be $-315-360n$, where $n$ is a natural number, and $-\frac{7\pi}{4}-2n\pi$, where $n$ is again a natural number.

Therefore, the two formulae for degrees are $45+360n$ and $-315-360n$, where $n$ is a natural number. These can be combined to $45+360z$, where $z$ is an integer.

Similarly, the two formulae for radians can be combined to $\frac{\pi}{4}+2z\pi$, where $z$ is an integer.