# Coterminal Angles Calculator | Formulas (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 $[0, 360\degree)$[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.

## 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:

(Video) Coterminal Angles

$\beta = \alpha \pm(2\pi\times k)$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 $0\degree$ 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 $0\degree \leq \theta < 360\degree$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:

(Video) Coterminal Angle Calculator

• 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 $0\degree$ 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$0\degree$ 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=α360°×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 $0\degree$: $360\degree$360°, $720\degree$720°, $-360\degree$360°, $-720\degree$720°.

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

(Video) Coterminal Angles - Positive and Negative, Converting Degrees to Radians, Unit Circle, Trigonometry

• Coterminal angle of $5\degree$: $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) Coterminal Angles | Basic Introduction | Sample Problems | Trigonometry | Pre-Calculus

• Coterminal 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): $0\degree$, $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.

(Video) FINDING COTERMINAL ANGLES || TRIGONOMETRY

## FAQs

### What is the easiest way to find Coterminal angles? ›

Finding coterminal angles is as simple as adding or subtracting 360° or 2π to each angle, depending on whether the given angle is in degrees or radians. There are an infinite number of coterminal angles that can be found.

How do you calculate Coterminal angles? ›

We can find the coterminal angles of a given angle by using the following formula: Coterminal angles of a given angle θ may be obtained by either adding or subtracting a multiple of 360° or 2π radians. Coterminal of θ = θ + 360° × k if θ is given in degrees. Coterminal of θ = θ + 2π × k if θ is given in radians.

What is the Coterminal angle of 720? ›

Subtract 360° 360 ° from 720° 720 ° . The resulting angle of 360° 360 ° is positive, less than 360° 360 ° , and coterminal with 720° 720 ° .

How many angles are Coterminal to if 45 degrees? ›

In Mathematics, the coterminal angle is defined as an angle, where two angles are drawn in the standard position. Also both have their terminal sides in the same location. For example, the coterminal angle of 45 is 405 and -315. Here 405 is the positive coterminal angle, -315 is the negative coterminal angle.

What is the Coterminal angle of 120? ›

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

What is the Coterminal angle of 750? ›

Subtract 360° 360 ° from 750° 750 ° . The resulting angle of 390° 390 ° is positive and coterminal with 750° 750 ° but isn't less than 360° 360 ° .

What is the Coterminal angle of 540? ›

540∘ is coterminal with 180∘, so I would just say it's "coterminal with a straight angle".

What are 5 examples of Coterminal angles? ›

In the above figure, 45°, 405° and -315° are coterminal angles having the same initial side (x-axis) and the same terminal side but with different amount of rotations. Other Examples: Similarly, 30°, -330°, 390° and 57°, 417°, -303° are also coterminal angles.

How do you calculate positive and negative Coterminal angles? ›

To find a positive and a negative angle coterminal with a given angle, you can add and subtract 360°if the angle is measured in degrees or 2π if the angle is measured in radians. Example 1: Find a positive and a negative angle coterminal with a 55°angle. A −305°angle and a 415°angle are coterminal with a 55°angle.

What is the Coterminal angle of 560? ›

since 560° is already greater than 360° , we can not add 360° but we need to start subtracting 360° . since this is not within our domain, the only coterminal angles of 560° are 200° and 160 - ° .

### What is the Coterminal angle of 765? ›

To find the value of cot 765 degrees using the unit circle, represent 765° in the form (2 × 360°) + 45° [∵ 765°>360°] ∵ The angle 765° is coterminal to 45° angle and also cotangent is a periodic function, cot 765° = cot 45°.

What is the Coterminal angle of 470? ›

Subtract 360° 360 ° from 470° 470 ° . The resulting angle of 110° 110 ° is positive, less than 360° 360 ° , and coterminal with 470° 470 ° .

What angle is Coterminal to 30 degrees? ›

For example, the angles 30°, –330° and 390° are all coterminal (see figure 2.1 below).

What angle is Coterminal to 145 degrees? ›

The angles which will be coterminal with 145° angle will be -215°, 505°, -575°, 865°, -1135° ……. The angles coterminal with 145° will be an infinite list. We add or subtract 360° to the given angle to get the coterminal angles. We add or subtract multiples of 360° to find all the coterminal angles.

Which angle is Coterminal to 128? ›

All angles having a measure of 128° + 360k°, where k is an integer, are coterminal with 128°. A positive angle is 28° + 360°(3) or 1208°.

What is the Coterminal angle of 170? ›

Add 360° 360 ° to −170° - 170 ° . The resulting angle of 190° 190 ° is positive and coterminal with −170° - 170 ° .

What angle is Coterminal to 110? ›

Answer and Explanation: We are given an angle that is -110 degrees. To find the coterminal angle of -110 degrees that is between 0 and 360 degrees, we could add 360 degrees to -110. The coterminal angle of -110 degrees that is in between 0 degree and 360 degrees is 250 degrees.

What angle is Coterminal with 130? ›

Add 360° 360 ° to −130° - 130 ° . The resulting angle of 230° 230 ° is positive and coterminal with −130° - 130 ° .

What is the Coterminal angle of 800? ›

An angle of 800 degrees is coterminal with an angle of 800- 360 = 440 degrees. It would also be coterminal with an angle of 440-360 = 80 degrees.

What is the Coterminal angle of 550? ›

Subtract 360° 360 ° from 550° 550 ° . The resulting angle of 190° 190 ° is positive, less than 360° 360 ° , and coterminal with 550° 550 ° .

### What is the Coterminal of 500? ›

Subtract 360° 360 ° from 500° 500 ° . The resulting angle of 140° 140 ° is positive, less than 360° 360 ° , and coterminal with 500° 500 ° .

What is the Coterminal angle of 515? ›

Subtract 360° 360 ° from 515° 515 ° . The resulting angle of 155° 155 ° is positive, less than 360° 360 ° , and coterminal with 515° 515 ° .

What is the Coterminal angle of 585? ›

Explanation: For cos 585°, the angle 585° > 360°. Given the periodic property of the cosine function, we can represent it as cos(585° mod 360°) = cos(225°). The angle 585°, coterminal to angle 225°, is located in the Third Quadrant(Quadrant III).

Are 135 and 495 Coterminal angles? ›

The coterminal angle is 495° − 360° = 135°.

What is the Coterminal angle of 330? ›

Add 360° 360 ° to −330° - 330 ° . The resulting angle of 30° 30 ° is positive and coterminal with −330° - 330 ° . Since 30° is in the first quadrant, the reference angle is 30° .

Are Coterminal angles always positive? ›

Coterminal angles can be positive and negative and involve rotations of multiples of 360 degrees! In fact, coterminal angles allow us to have infinite representations of angles in standard position with the same terminal side.

What angle is Coterminal with degrees? ›

Coterminal angles have the same initial and terminal sides. The simplest case is 180°. If you imagine this on a cartesian plane, it is simply the x-axis. The 180° on the positive y-axis side is coterminal with the 180° on the negative y-axis side and vice versa.

How to find a positive angle less than 2pi that is coterminal with the given angle? ›

In order to find a positive angle that is coterminal with the given angle which is less than 2π , we subtract 2π and apply two full rotations (n=2) to the given angle. Hence, a positive angle less than 2π that is coterminal with the given angle is 4π3 4 π 3 .

What is the positive Coterminal of 450? ›

Therefore the positive and negative coterminal angles of 450∘ are 90∘ and −90∘ , respectively.

What is the Coterminal angle of 1380? ›

Subtract 360° 360 ° from 1380° 1380 ° . The resulting angle of 1020° 1020 ° is positive and coterminal with 1380° 1380 ° but isn't less than 360° 360 ° .

### What is the Coterminal angle of 870? ›

Subtract 360° 360 ° from 870° 870 ° . The resulting angle of 510° 510 ° is positive and coterminal with 870° 870 ° but isn't less than 360° 360 ° .

What is the Coterminal angle of 580? ›

Subtract 360° 360 ° from 580° 580 ° . The resulting angle of 220° 220 ° is positive, less than 360° 360 ° , and coterminal with 580° 580 ° .

What angles are Coterminal with 250? ›

Trigonometry Examples

The resulting angle of 110° 110 ° is positive and coterminal with −250° - 250 ° .

What is the Coterminal angle of 310? ›

Add 360° 360 ° to −310° - 310 ° . The resulting angle of 50° 50 ° is positive and coterminal with −310° - 310 ° .

What angles are Coterminal with 480? ›

Subtract 360° 360 ° from 480° 480 ° . The resulting angle of 120° 120 ° is positive, less than 360° 360 ° , and coterminal with 480° 480 ° .

Which angle is Coterminal 160? ›

Add 360° 360 ° to −160° - 160 ° . The resulting angle of 200° 200 ° is positive and coterminal with −160° - 160 ° .

What angle is Coterminal to 510 degrees? ›

Subtract 360° 360 ° from 510° 510 ° . The resulting angle of 150° 150 ° is positive, less than 360° 360 ° , and coterminal with 510° 510 ° .

What angle is Coterminal to 20 degrees? ›

This means that the first positive coterminal angle to 20 degrees is 380 degrees, followed by 740 degrees, 1100 degrees, 1460 degrees, and so on.

Is 750 degrees Coterminal with 30 degrees? ›

The resulting angle of 30° 30 ° is positive, less than 360° 360 ° , and coterminal with 750° 750 ° .

What angles are Coterminal with 135? ›

Trigonometry Examples

Find an angle that is positive, less than 360° , and coterminal with −135° . Add 360° 360 ° to −135° - 135 ° . The resulting angle of 225° 225 ° is positive and coterminal with −135° - 135 ° .

### What is the Coterminal angle of 158? ›

The resulting angle of 158° 158 ° is positive and coterminal with −202° - 202 ° . Since the angle 158° is in the second quadrant, subtract 158° from 180° .

What Coterminal is 150? ›

Coterminal presumably refers to something like the same spot on the unit circle. That means the angles differ by a multiple of 360∘ or of 2π radians. So a positive angle coterminal with −150∘ would be −150∘+360∘=210∘.

What is the Coterminal angle of 320? ›

Add 360° 360 ° to −320° - 320 ° . The resulting angle of 40° 40 ° is positive and coterminal with −320° - 320 ° . Since 40° is in the first quadrant, the reference angle is 40° .

What is the Coterminal of 115? ›

The coterminal angles of 115∘ are 475∘ and −245∘ .

What is the Coterminal of 125? ›

Add 360° 360 ° to −125° - 125 ° . The resulting angle of 235° 235 ° is positive and coterminal with −125° - 125 ° .

What angle is Coterminal with 810 degrees? ›

Trigonometry Examples

Subtract 360° 360 ° from 810° 810 ° . The resulting angle of 450° 450 ° is positive and coterminal with 810° 810 ° but isn't less than 360° 360 ° .

How do you find the positive and negative Coterminal angles? ›

To find a positive and a negative angle coterminal with a given angle, you can add and subtract 360°if the angle is measured in degrees or 2π if the angle is measured in radians. Example 1: Find a positive and a negative angle coterminal with a 55°angle. A −305°angle and a 415°angle are coterminal with a 55°angle.

What is the Coterminal angle of 760? ›

t = 760 ∘ . The angle is greater than two times 360∘ . So, we have to subtract 720∘ from the given angle t to obtain the coterminal angle θ . Hence, the coterminal angle is 40∘ .

What is the Coterminal angle of 420? ›

Subtract 360° 360 ° from 420° 420 ° . The resulting angle of 60° 60 ° is positive, less than 360° 360 ° , and coterminal with 420° 420 ° .

What angle between 0 and 360 is Coterminal with 600? ›

Summary: The angle between 0 degrees and 360 degrees that is coterminal with -600 degrees -240 degrees and 120 degrees b.

### What angle is more than 180 but less than 360? ›

An angle which measures more than 180∘ but less than 360∘ is called an obtuse angle.

What is the Coterminal angle of 460? ›

Coterminal angles can be found either by subtracting or adding 360∘ to the given angle. So 460∘ is a positive coterminal angle of 820∘ .

What angle between 0 and 360 is coterminal with 440? ›

The resulting angle of 80° 80 ° is positive, less than 360° 360 ° , and coterminal with 440° 440 ° .

What is the angle between 0 and 360 that is Coterminal with 375? ›

To find coterminal angles, we have to either add or subtract 360∘ . However, adding 360∘ would result to a value outside the domain that we are considering. So, we simply subtract 360∘ from 375∘ . Therefore, the required angle is 15∘ .

What angle between 0 and 360 is coterminal with 810? ›

The resulting angle of 90° 90 ° is positive, less than 360° 360 ° , and coterminal with 810° 810 ° .

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