# Coterminal Angle Calculator

Created by Hanna Pamuła, PhD
Reviewed by Bogna Szyk and Jack Bowater
Last updated: Sep 28, 2023

Welcome to our coterminal angle calculator – a tool that will solve many of your problems regarding coterminal angles:

• Do you want to find a coterminal angle of a given angle, preferably in the $[0, 360\degree)$ range? Great news: you can see here.
• Are you hunting for positive and negative coterminal angles? Also here.
• Would you like to check if two angles are coterminal? Check! ✔️
• Are you searching for a coterminal angles calculator for radians? Good for you, our tool works both for π radians and degrees.
• Or maybe you're looking for a coterminal angles definition, with some examples? Then you won't be disappointed with this calculator.
• Will the tool guarantee me a passing grade on my math quiz? ❌ Well, our tool is versatile, but that's on you :)

Use our calculator to solve your coterminal angles issues, or scroll down to read more.

## What is a coterminal angle?

Coterminal angles are those angles that share the terminal side of an angle occupying the standard position. The standard position means that one side of the angle is fixed along the positive x-axis, and the vertex is located at the origin.

In other words, two angles are coterminal when the angles themselves are different, but their sides and vertices are identical. Also, you can remember the definition of the coterminal angle as angles that differ by a whole number of complete circles.

Look at the picture below, and everything should be clear! So, as we said: all the coterminal angles start at the same side (initial side) and share the terminal side.

The thing which can sometimes be confusing is the difference between the reference angle and coterminal angles definitions. Remember that they are not the same thing – the reference angle is the angle between the terminal side of the angle and the x-axis, and it's always in the range of $[0, 90\degree]$ (or $[0, \pi/2]$): for more insight on the topic, visit our reference angle calculator!

## How to find coterminal angles? Coterminal angles formula

To find the coterminal angles to your given angle, you need to add or subtract a multiple of 360° (or 2π if you're working in radians). So, to check whether the angles α and β are coterminal, check if they agree with a coterminal angles formula:

a) For angles measured in degrees:

$\beta=\alpha\pm(360\degree\times k)$

where $k$ is a positive integer.

b) For angles measured in radians:

$\beta = \alpha \pm(2\pi\times k)$

here k is a positive integer

A useful feature is that in trigonometry functions calculations, any two coterminal angles have exactly the same trigonometric values. So if $\beta$ and $\alpha$ are coterminal, then their sines, cosines and tangents are all equal.

When calculating the sine, for example, we say:

$\sin(\alpha) = \sin(\alpha\pm(360\degree \times k))$

## How to find a coterminal angle between 0 and 360° (or 0 and 2π)?

To determine the coterminal angle between $0\degree$ and $360\degree$, all you need to do is to calculate the modulo – in other words, divide your given angle by the $360\degree$ and check what the remainder is.

We'll show you how it works with two examples – covering both positive and negative angles. We want to find a coterminal angle with a measure of $\theta$ such that $0\degree \leq \theta < 360\degree$, for a given angle equal to:

$420\degree\text{mod}\ 360\degree = 60\degree$

How to do it manually?

• First, divide one number by the other, rounding down (we calculate the floor function): $\left\lfloor420\degree/360\degree\right\rfloor = 1$.

• Then, multiply the divisor by the obtained number (called the quotient): $360\degree \times 1 = 360\degree$.

• Subtract this number from your initial number: $420\degree - 360\degree = 60\degree$.

Substituting these angles into the coterminal angles formula gives $420\degree = 60\degree + 360\degree\times 1$.

#### -858°

$-858\degree \text{mod}\ 360\degree = 222\degree$

Repeating the steps from above:

• Calculate the floor: $\left\lfloor-858\degree / 360\degree\right\rfloor = -3$.
• Find the total full circles: $360\degree \times -3 = -1080\degree$.
• Calculate the remainder: $-858\degree + 1080\degree = 222\degree$.

So the coterminal angles formula, $\beta = \alpha \pm 360\degree \times k$, will look like this for our negative angle example:

$-858\degree = 222\degree - 360\degree\times 3$

The same works for the $[0,2\pi)$ range, all you need to change is the divisor – instead of $360\degree$, use $2\pi$.

Now, check the results with our coterminal angle calculator – it displays the coterminal angle between $0\degree$ and $360\degree$ (or $0$ and $2\pi$), as well as some exemplary positive and negative coterminal angles.

## Positive and negative coterminal angles

If you want to find a few positive and negative coterminal angles, you need to subtract or add a number of complete circles. But how many?

One method is to find the coterminal angle in the$0\degree$ and $360\degree$ range (or $[0,2\pi)$ range), as we did in the previous paragraph (if your angle is already in that range, you don't need to do this step). Then just add or subtract $360\degree$, $720\degree$, $1080\degree$... ($2\pi$,$4\pi$,$6\pi$...), to obtain positive or negative coterminal angles to your given angle.

For example, if $\alpha = 1400\degree$, then the coterminal angle in the $[0,360\degree)$ range is $320\degree$ – which is already one example of a positive coterminal angle.

• Other positive coterminal angles are $680\degree$, $1040\degree$...

• Other negative coterminal angles are $-40\degree$, $-400\degree$, $-760\degree$...

Also, you can simply add and subtract a number of revolutions if all you need is any positive and negative coterminal angle. For our previously chosen angle, $\alpha = 1400\degree$, let's add and subtract $10$ revolutions (or $100$, why not):

• Positive coterminal angle: $\beta = \alpha + 360\degree \times 10 = 1400\degree + 3600\degree = 5000\degree$.

• Negative coterminal angle: $\beta = \alpha - 360\degree\times 10 = 1400\degree - 3600\degree = -2200\degree$.

The number or revolutions must be large enough to change the sign when adding/subtracting. For example, one revolution for our exemplary α is not enough to have both a positive and negative coterminal angle – we'll get two positive ones, $1040\degree$ and $1760\degree$.

## What is a coterminal angle of...

If you're wondering what the coterminal angle of some angle is, don't hesitate to use our tool – it's here to help you!

But if, for some reason, you still prefer a list of exemplary coterminal angles (but we really don't understand why...), here you are:

• Coterminal angle of $0\degree$: $360\degree$, $720\degree$, $-360\degree$, $-720\degree$.

• Coterminal angle of $1\degree$: $361\degree$, $721\degree$ $-359\degree$, $-719\degree$.

• Coterminal angle of $5\degree$: $365\degree$, $725\degree$, $-355\degree$, $-715\degree$.

• Coterminal angle of $10\degree$: $370\degree$, $730\degree$, $-350\degree$, $-710\degree$.

• Coterminal angle of $15\degree$: $375\degree$, $735\degree$, $-345\degree$, $-705\degree$.

• Coterminal angle of $20\degree$: $380\degree$, $740\degree$, $-340\degree$, $-700\degree$.

• Coterminal angle of $25\degree$: $385\degree$, $745\degree$, $-335\degree$, $-695\degree$.

• Coterminal angle of $30\degree$ ($\pi / 6$): $390\degree$, $750\degree$, $-330\degree$, $-690\degree$.

• Coterminal angle of $45\degree$ ($\pi / 4$): $495\degree$, $765\degree$, $-315\degree$, $-675\degree$.

• Coterminal angle of $60\degree$ ($\pi / 3$): $420\degree$, $780\degree$, $-300\degree$, $-660\degree$

• Coterminal angle of $75\degree$: $435\degree$, $795\degree$,$-285\degree$, $-645\degree$

• Coterminal angle of $90\degree$ ($\pi / 2$): $450\degree$, $810\degree$, $-270\degree$, $-630\degree$.

• Coterminal angle of $105\degree$: $465\degree$, $825\degree$,$-255\degree$, $-615\degree$.

• Coterminal angle of $120\degree$ ($2\pi/ 3$): $480\degree$, $840\degree$, $-240\degree$, $-600\degree$.

• Coterminal angle of $135\degree$ ($3\pi / 4$): $495\degree$, $855\degree$, $-225\degree$, $-585\degree$.

• Coterminal angle of $150\degree$ ($5\pi/ 6$): $510\degree$, $870\degree$, $-210\degree$, $-570\degree$.

• Coterminal angle of $165\degree$: $525\degree$, $885\degree$, $-195\degree$, $-555\degree$.

• Coterminal angle of $180\degree$ ($\pi$): $540\degree$, $900\degree$, $-180\degree$, $-540\degree$

• Coterminal angle of $195\degree$: $555\degree$, $915\degree$, $-165\degree$, $-525\degree$.

• Coterminal angle of $210\degree$ ($7\pi / 6$): $570\degree$, $930\degree$, $-150\degree$, $-510\degree$.

• Coterminal angle of $225\degree$ ($5\pi / 4$): $585\degree$, $945\degree$, $-135\degree$, $-495\degree$.

• Coterminal angle of $240\degree$ ($4\pi / 3$: $600\degree$, $960\degree$, $120\degree$, $-480\degree$.

• Coterminal angle of $255\degree$: $615\degree$, $975\degree$, $-105\degree$, $-465\degree$.

• Coterminal angle of $270\degree$ ($3\pi / 2$): $630\degree$, $990\degree$, $-90\degree$, $-450\degree$.

• Coterminal angle of $285\degree$: $645\degree$, $1005\degree$, $-75\degree$, $-435\degree$.

• Coterminal angle of $300\degree$ ($5\pi / 3$): $660\degree$, $1020\degree$, $-60\degree$, $-420\degree$.

• Coterminal angle of $315\degree$ ($7\pi / 4$): $675\degree$, $1035\degree$, $-45\degree$, $-405\degree$.

• Coterminal angle of $330\degree$ ($11\pi / 6$): $690\degree$, $1050\degree$, $-30\degree$, $-390\degree$.

• Coterminal angle of $345\degree$: $705\degree$, $1065\degree$, $-15\degree$, $-375\degree$.

• Coterminal angle of $360\degree$ ($2\pi$): $0\degree$, $720\degree$, $-360\degree$, $-720\degree$.

If you didn't find your query on that list, type the angle into our coterminal angle calculator – you'll get the answer in the blink of an eye!

## FAQ

### What is the coterminal angle of 1000° between 0° and 360°?

The answer is 280°. To arrive at this result, recall the formula for coterminal angles of 1000°:

1000° + 360° × k.

Clearly, to get a coterminal angle between 0° and 360°, we need to use negative values of k. For k=-1, we get 640°, which is too much. So let's try k=-2: we get 280°, which is between 0° and 360°, so we've got our answer.

### How do I find all coterminal angles?

A given angle has infinitely many coterminal angles, so you cannot list all of them. You can write them down with the help of a formula. If your angle θ is expressed in degrees, then the coterminal angles are of the form θ + 360° × k, where k is an integer (maybe a negative number!). If θ is in radians, then the formula reads θ + 2π × k.

### What are the coterminal angles of 45°?

The coterminal angles of 45° are of the form 45° + 360° × k, where k is an integer. Plugging in different values of k, we obtain different coterminal angles of 45°. Let us list several of them:

45°, 405°, 765°, -315°, -675°.

### How do I check if two angles are coterminal?

Two angles, α and β, are coterminal if their difference is a multiple of 360°. That is, if β - α = 360° × k for some integer k.

For instance, the angles -170° and 550° are coterminal, because 550° - (-170°) = 720° = 360° × 2. If your angles are expressed in radians instead of degrees, then you look for multiples of 2π, i.e., the formula is β - α = 2π × k for some integer k.

Hanna Pamuła, PhD
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