Coterminal Angles: How to Find Them

Coterminal angles are angles in standard position that share the same terminal side. To find an angle coterminal with $\theta$ , add or subtract 360°, one full turn. Every angle has infinitely many coterminal angles, given by $\theta + 360^\circ n$ , where $n$ is any integer.

Before this lesson: You should know Angles in Standard Position and Adding and Subtracting Integers.

Same Ending, Different Turns

Picture the terminal side of a 50° angle. Now turn one more full circle counterclockwise. You end on the same ray, but the angle is now $50^\circ + 360^\circ = 410^\circ$ . If you start over and turn clockwise instead until you reach that ray, you turn $50^\circ - 360^\circ = -310^\circ$ .

All three angles, 50°, 410° and −310°, end on the same terminal side. That is what coterminal means: “same terminal side.” They have the same trig values, which is why this idea matters in trigonometry.

An angle of 50 degrees in standard position with three arcs ending on the same terminal side. A small blue arc turns 50 degrees counterclockwise. A green arc turns one full circle and then 50 degrees more, for 410 degrees. An orange arc turns 310 degrees clockwise, for −310 degrees. — The angles 50°, 410° and −310° are coterminal. Each one ends on the same terminal side.

The Coterminal Angle Formula

\theta + 360^\circ n, \quad n = \ldots, -2, -1, 0, 1, 2, \ldots

Each value of the integer $n$ gives a different coterminal angle. A positive $n$ adds full turns, and a negative $n$ subtracts them. Two angles are coterminal exactly when their difference is a multiple of 360°.

To find a coterminal angle, add or subtract 360°.
To find the coterminal angle between 0° and 360°, divide the angle by 360 to see how many full turns it holds.
Subtract that many full turns (or add them, for a negative angle) until the result is between 0° and 360°.
To test whether two angles are coterminal, subtract them and check whether the difference is a multiple of 360°.

Worked Examples

Example 1: One positive and one negative coterminal angle

Find a positive angle and a negative angle that are coterminal with $50^\circ$ .

50^\circ + 360^\circ = 410^\circ

Add one full turn.

50^\circ - 360^\circ = -310^\circ

Subtract one full turn.

Answer: $410^\circ$ and $-310^\circ$ . Other correct answers include $770^\circ$ and $-670^\circ$ .

Example 2: Reduce a large angle

Find the angle between 0° and 360° that is coterminal with $820^\circ$ .

Divide by 360 to count the full turns: $\frac{820}{360} \approx 2.28$ , so there are 2 full turns

2 \cdot 360^\circ = 720^\circ

Find 2 full turns in degrees.

820^\circ - 720^\circ = 100^\circ

Subtract them.

Answer: $100^\circ$ . It is in Quadrant II.

Example 3: A negative angle

Find the angle between 0° and 360° that is coterminal with $-530^\circ$ .

-530^\circ + 360^\circ = -170^\circ

Add one full turn. It is still negative.

-170^\circ + 360^\circ = 190^\circ

Add another full turn.

Answer: $190^\circ$ . You added $2 \cdot 360^\circ = 720^\circ$ in all.

Example 4: Test two angles

Are $145^\circ$ and $-935^\circ$ coterminal?

145^\circ - (-935^\circ) = 145^\circ + 935^\circ = 1080^\circ

Subtract the angles (see adding and subtracting integers)

\frac{1080}{360} = 3

Divide the difference by 360.

The result is a whole number, so the difference is 3 full turns.

Answer: Yes, they are coterminal.

Example 5: Word problem with a Ferris wheel

A Ferris wheel seat starts level with the center, on the right side. The wheel turns counterclockwise through $1170^\circ$ . Where is the seat when the wheel stops?

Put the center at the origin, so the seat starts on the positive x-axis.

Divide by 360 to count the full turns: $\frac{1170}{360} = 3.25$ , so there are 3 full turns

3 \cdot 360^\circ = 1080^\circ

Find 3 full turns in degrees.

1170^\circ - 1080^\circ = 90^\circ

Subtract them.

An angle of 90° ends on the positive y-axis.

Answer: The seat is at the very top of the wheel, after three and a quarter turns.

Common Mistakes

Mistake	Example of the error	Fix
Adding 180° instead of 360°	Saying 50° and 230° are coterminal	A full turn is 360°. 230° points the opposite way.
Stopping too early with a negative angle	Giving −170° as the answer between 0° and 360° for −530°	Keep adding 360° until the angle is between 0° and 360°: 190°.
Using the decimal part of the division as degrees	$\frac{820}{360} \approx 2.28$ , so the answer is 28°	Use only the whole number of turns, then subtract: $820^\circ - 720^\circ = 100^\circ$ .
Subtracting a negative wrong	$145^\circ - (-935^\circ) = -790^\circ$	Subtracting a negative means adding: $145^\circ + 935^\circ = 1080^\circ$ .
Thinking an angle has only one coterminal angle	Listing only 410° for 50°	Any number of full turns works, so there are infinitely many.

Related Methods

Coterminal angles in radians

One full turn is also $2\pi$ radians, as the next lesson shows. In radians, coterminal angles are $\theta + 2\pi n$ . For example, $\frac{\pi}{4}$ and $\frac{9\pi}{4}$ are coterminal.

Using the remainder

For a positive angle, the coterminal angle between 0° and 360° is the remainder when you divide by 360. For 820°, $820 = 2 \cdot 360 + 100$ , so the remainder is 100°.

Why it matters later

Coterminal angles have the same sine, cosine and tangent. In trigonometric ratios you used angles less than 90°. Later in this course you will reduce any angle to one between 0° and 360° before finding its trig values.

Practice Problems

Find one positive and one negative angle coterminal with $30^\circ$ .
Find one positive and one negative angle coterminal with $200^\circ$ .
Find one positive and one negative angle coterminal with $-75^\circ$ .
Find the angle between 0° and 360° coterminal with $765^\circ$ .
Find the angle between 0° and 360° coterminal with $1000^\circ$ .
Find the angle between 0° and 360° coterminal with $-100^\circ$ .
Find the angle between 0° and 360° coterminal with $-800^\circ$ .
Are $60^\circ$ and $780^\circ$ coterminal?
Are $25^\circ$ and $-385^\circ$ coterminal?
A bicycle wheel turns clockwise through $-1350^\circ$ . Find the angle between 0° and 360° that is coterminal with this rotation.

Show answers

30^\circ + 360^\circ = 390^\circ

Add 360°.

30^\circ - 360^\circ = -330^\circ

Subtract 360°.

Answer: $390^\circ$ and $-330^\circ$

200^\circ + 360^\circ = 560^\circ

Add 360°.

200^\circ - 360^\circ = -160^\circ

Subtract 360°.

Answer: $560^\circ$ and $-160^\circ$

-75^\circ + 360^\circ = 285^\circ

Add 360°.

-75^\circ - 360^\circ = -435^\circ

Subtract 360°.

Answer: $285^\circ$ and $-435^\circ$

Count the full turns: $\frac{765}{360} \approx 2.13$ , so 2 turns

2 \cdot 360^\circ = 720^\circ

Find 2 turns.

765^\circ - 720^\circ = 45^\circ

Subtract.

Answer: $45^\circ$

Count the full turns: $\frac{1000}{360} \approx 2.78$ , so 2 turns

2 \cdot 360^\circ = 720^\circ

Find 2 turns.

1000^\circ - 720^\circ = 280^\circ

Subtract.

Answer: $280^\circ$

-100^\circ + 360^\circ = 260^\circ

Add 360°.

Answer: $260^\circ$

-800^\circ + 360^\circ = -440^\circ

Add 360°.

-440^\circ + 360^\circ = -80^\circ

Add 360° again.

-80^\circ + 360^\circ = 280^\circ

Add 360° again.

Answer: $280^\circ$

780^\circ - 60^\circ = 720^\circ

Subtract.

\frac{720}{360} = 2

Divide by 360 (a whole number).

Answer: yes, they are coterminal

25^\circ - (-385^\circ) = 25^\circ + 385^\circ = 410^\circ

Subtract.

\frac{410}{360} \approx 1.14

Divide by 360 (not a whole number).

Answer: no, they are not coterminal

10.

Count the full turns: $\frac{1350}{360} = 3.75$ , so add 4 turns to get a positive angle

4 \cdot 360^\circ = 1440^\circ

Find 4 turns.

-1350^\circ + 1440^\circ = 90^\circ

Add.

Answer: $90^\circ$

FAQ

What are coterminal angles?

They are angles in standard position that end on the same terminal side. For example, 30°, 390° and −330° are all coterminal.

How do you find a coterminal angle?

Add or subtract 360° as many times as you like. In radians, add or subtract $2\pi$ .

How many coterminal angles does an angle have?

Infinitely many. Every whole number of full turns, forward or backward, gives another one.

How do you know if two angles are coterminal?

Subtract them. If the difference is a multiple of 360°, such as 0°, 360° or −720°, they are coterminal.

Summary

Coterminal angles share the same terminal side.
Add or subtract 360° to find one: $\theta + 360^\circ n$ .
Every angle has infinitely many coterminal angles.
To reduce an angle, remove whole turns until it is between 0° and 360°.
Two angles are coterminal when their difference is a multiple of 360°.

Previous lesson: Angles in Standard Position

Next lesson: Radians

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