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 $[0, 360\degree)$[0,360°) 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.

Let's start with the coterminal angles definition.

## 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 $[0, 90\degree]$[0,90°] (or $[0, \pi/2]$[0,π/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:

$\beta=\alpha\pm(360\degree\times k)$β=α±(360°×k)

where $k$k is a positive integer.

b) For angles measured in radians:

$\beta = \alpha \pm(2\pi\times k)$β=α±(2π×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 $\beta$β and $\alpha$α are coterminal, then their sines, cosines and tangents are all equal.

When calculating the sine, for example, we say:

$\sin(\alpha) = \sin(\alpha\pm(360\degree \times k))$sin(α)=sin(α±(360°×k))

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

To determine the coterminal angle between $0\degree$0° and $360\degree$360°, all you need to do is to calculate the modulo – in other words, divide your given angle by the $360\degree$360° 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 $\theta$θ such that $0\degree \leq \theta < 360\degree$0°≤θ<360°, for a given angle equal to:

$420\degree\text{mod}\ 360\degree = 60\degree$420°mod360°=60°

How to do it manually?

First, divide one number by the other, rounding down (we calculate the floor function): $\left\lfloor420\degree/360\degree\right\rfloor = 1$⌊420°/360°⌋=1.

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

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

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

#### -858°

$-858\degree \text{mod}\ 360\degree = 222\degree$−858°mod360°=222°

Repeating the steps from above:

- Calculate the floor: $\left\lfloor-858\degree / 360\degree\right\rfloor = -3$⌊−858°/360°⌋=−3.
- Find the total full circles: $360\degree \times -3 = -1080\degree$360°×−3=−1080°.
- Calcualte teh remainder: $-858\degree + 1080\degree = 222\degree$−858°+1080°=222°.

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

$-858\degree = 222\degree - 360\degree\times 3$−858°=222°−360°×3

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

Now, check the results with our **coterminal angle calculator** – it displays the coterminal angle between $0\degree$0° and $360\degree$360° (or $0$0 and $2\pi$2π), 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$0° and $360\degree$360° range (or $[0,2\pi)$[0,2π) 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$360°, $720\degree$720°, $1080\degree$1080°... ($2\pi$2π,$4\pi$4π,$6\pi$6π...), to obtain positive or negative coterminal angles to your given angle.

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

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

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

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, $\alpha = 1400\degree$α=1400°, let's add and subtract $10$10 revolutions (or $100$100, why not):

Positive coterminal angle: $\beta = \alpha + 360\degree \times 10 = 1400\degree + 3600\degree = 5000\degree$β=α+360°×10=1400°+3600°=5000°.

Negative coterminal angle: $\beta = \alpha - 360\degree\times 10 = 1400\degree - 3600\degree = -2200\degree$β=α−360°×10=1400°−3600°=−2200°.

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$1040° and $1760\degree$1760°.

## 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$0°: $360\degree$360°, $720\degree$720°, $-360\degree$−360°, $-720\degree$−720°.

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

(Video) Coterminal Angles - Positive and Negative, Converting Degrees to Radians, Unit Circle, TrigonometryCoterminal angle of $5\degree$5°: $365\degree$365°, $725\degree$725°, $-355\degree$−355°, $-715\degree$−715°.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Coterminal angle of $180\degree$180° ($\pi$π): $540\degree$540°, $900\degree$900°, $-180\degree$−180°, $-540\degree$−540°

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

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

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

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

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

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

(Video) Determining coterminal angles by adding and subtracting 2piCoterminal angle of $285\degree$285°: $645\degree$645°, $1005\degree$1005°, $-75\degree$−75°, $-435\degree$−435°.

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

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

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

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

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

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`

.