Coterminal Angle Calculator Formula, Solved Examples, Step by Step

So, before we go into the specifics of what a coterminal angle calculator is and how to use one, let’s review what coterminal angles are. As a result, coterminal angles are angles that have the same terminal side as an angle in the usual position. As a result, the following question is: what is the accepted position? One side of the angle is fixed along the positive x-axis in this case. The vertex, on the other hand, is located at the origin.

As a result, coterminal angles are those that are distinct yet have the same sides and vertices. Furthermore, you can define them as angles that are separated by a large number of complete circles. As a result, all coterminal angles start on the same side, the beginning side. They do, however, share the terminal side.

However, there is one point that is frequently misunderstood. This is the discrepancy between the definition of the coterminal angles and the primary reference angle. So, keep in mind that these are two very different things. The reference angle is the angle formed by the angle’s terminal side and the x-axis, and it is always between [0, 90] (or [0, /2]).

Coterminal Angle Calculator

The Coterminal Angles Calculator is a web-based application for calculating the positive and negative coterminal angles of a given angle. Angles with the identical initial and terminal sides are referred to as coterminal angles. It aids in the computation of the positive and negative coterminal angles of a given angle in degrees. We add or subtract 360 degrees from a coterminal angle to ascertain it. Enter the angle in the supplied input box to utilise the coterminal angles calculator.

Coterminal Angle Calculator With Steps

As a result, everything has gotten exceedingly simple in the age of the internet. You’re having trouble grasping a math concept? Check it out on the internet! Calculating coterminal angles, which used to be a difficult task, has now become a breeze.

If you don’t care about the specific coterminal angle calculation, you may just obtain the value from a coterminal angle calculator and utilize it in a larger calculation. The coterminal angle calculator, on the other hand, can serve as an excellent validator for results if you’re only interested in the value. At the end of the day, we all make mistakes when performing manual computations. Furthermore, a minor error here and there can undo all of our hard work in a lengthy calculation.

There are a plethora of online coterminal angle calculators.

The following is how to use the coterminal angle calculator:

First step to use a coterminal angle calculator

So, you go to the website and look at the page. A full discussion of how to calculate coterminal angles can be found on the right-hand side. This will allow you to see whether you’re making any mistakes while manually performing it. The actual coterminal angle calculator is located in the upper left corner. If you want to get the right answer, you’ll need to fill in three boxes.

Second step to use a coterminal angle calculator

The first box reads “I wish to,” and there are two choices. As a result, you may either find the coterminal angles of a specific angle or determine whether two angles are coterminal. As a result, on the coterminal angle calculator, select the appropriate choice for your needs. “Angle in” is written in the next box. As a result, the computation might be done in degrees or radians. As a result, the coterminal angle calculator can calculate the value in two different methods. Choose the appropriate option.

Third step to use a coterminal angle calculator

There will be one more box if you wish to find a coterminal angle as a reference angle. “Angle,” it says. So, here’s where you’ll put the reference angle whose coterminal angles you’re looking for. The coterminal angle calculator will provide you with all of the positive values right away. However, there will be two boxes if you want to see if two angles are coterminal. “Angle 1” is written in one box, while “Angle 2” is written in the other. So you enter the two angles you want to verify into the coterminal angle calculator. The coterminal angle calculator asserts that if two angles are coterminal, they are coterminal. It states The two angles are not coterminal if they aren’t.

Fourth step to use a coterminal angle calculator

The calculation is now finished, and you have the result. As a result, you can also send the outcome. Just below the coterminal angle calculator is a “mail this result” button. When you click it, it generates a link that you can copy and share. You may also share it on social media sites like Facebook, Twitter, LinkedIn, and e-mail.

Coterminal Angle Calculator Formula

Finding coterminal angles can be difficult at times, which is why a coterminal angle calculator is necessary. However, throwing data in blindly to generate results is not a good idea. As a result, it’s a good idea to understand how the calculator works. This will also lessen your own errors, and you will be able to compare the two responses.

So, if you’re working with degrees, you’ll need to add or subtract a multiple of 360 degrees, and if you’re working with radians, you’ll need to add or subtract 2pi. Let’s pretend the two angles are a and b. So, you want to see if they’re coterminal now.

As a result, the coterminal angle calculator also employs two formulas. You’ll need to double-check the angles with one of them.

b = a 360 * k, where k is a positive integer, is a formula for measuring angles in degrees.

b = a 2 * k, where k is a positive integer, is a formula for measuring angles in radians.

Now, if you’re familiar with trigonometry, there’s a feature you should keep in mind when checking your coterminal angles. As a result, the trigonometric values of any two coterminal angles will be identical. So, what exactly does this imply? Assume that the angles a and b are coterminal. So, once we’ve discovered them, we can compute their sines, cosines, and tangents. As a result, the value for both will be the same for all of these functions. As a result, this is a simple approach to determine whether or not what you’re doing is correct. You may also use this method to double-check the answer you got with the coterminal angle calculator.

Coterminal Angle Calculator

Coterminal Angle Calculator with Radians

As we’ve seen, a coterminal angle calculator can be used to find coterminal angles in both radians and degrees. So, first, let’s look at how to get a radian value from a coterminal angle calculator.

Select the option in the CA calculator that indicates you want to find coterminal angles from a reference angle as the first step.

In the second step, select “radians” from the drop-down menu.

As a result, you already have a symbol in the next box. All you have to do is enter an integer. Let’s say you type in 3. As a result, the angle you’re examining is 3 pi.

It now appears to be a very complicated response. It also converts the angle to degrees. In this situation, the angle is 540 degrees. As a result, it has discovered a coterminal angle of 1 in the [0, 2] range, which is positioned on the positive x-axis. As a result, both positive and negative coterminal angles exist for this angle. Both may be found using the coterminal angle calculator. There are four positive coterminal angles in this situation. 3 pi, 5 pi, 7 pi, and 9 pi are the numbers. There are four negative coterminal angles, on the other hand.

-1 pi, -3 pi, -5 pi, and -7 pi are the values. (Remember that pi equals 180 degrees.) If you’re looking for a specific value, you may simply use the verify whether two angles are coterminal portions method. As a result, both values should be entered in degrees. If the angles are coterminal or not, the calculator will tell you. Let’s say the first angle is 45 degrees and the second is 90 degrees. As a result, they are not coterminal.

Coterminal Angle Calculator with Pi

However, there is one factor to keep in mind while determining whether two angles are coterminal. Remember that both angles must be entered into the coterminal angle calculator in the same unit. As a result, if you’re utilising degrees, you’ll need to use them in both circumstances. If you’re using pi radians, though, be sure you’re using pi in both circumstances. As a result, you shouldn’t expect the calculator to detect the unit conversion. Furthermore, it will not keep one intact, convert another, and compare the results. It can only make a comparison if the data is entered correctly.

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Furthermore, you have a strong understanding of the relationship between degrees and pi radians. Multiply the radians by 180° radians to convert from radians to degrees. As a result, pi denotes 180 degrees in both positive and negative directions. So, when you say 14 pi, you’re talking about 45 degrees. Now, if you enter 45 radians instead of 45 degrees in the calculation, the result will be incorrect. This is due to the fact that the value will now be 45 pi, which is 45 X 180 degrees, or 8100 degrees.

As a result, be extremely cautious with the units, as students frequently make mistakes here. The meaning varies when the unit is changed. This is also something that your coterminal angle calculator will be unable to identify. For a mechanical instrument, 8100 degrees is nothing out of the ordinary. As a result, it will simply review and provide you with the information you require.

Coterminal Angle Calculator Degrees Minutes Seconds

It’s possible that you won’t always represent angles as integers. So, you might have an angle value of x degrees, y minutes, and z seconds. As an angle, it’s valid. Your coterminal angle calculator, on the other hand, will not accept this. As a result, you must only enter integer values to obtain coterminal angles. The coterminal angle calculator works in the same way. You can, however, submit a value like 45.5 if you’re comparing it to another angle whose value you’ll also provide. However, there is no option to enter the value in minutes or seconds in any situation. To use your coterminal angle calculator, convert the minutes and seconds to degrees and enter the final value.

How Does a Coterminal Angle Calculator Work?

Coterminal angles are angles that occupy the same place despite having different values. Their vertices are the same, and they are in the same quadrant on the same side. When a defined angle is turned clockwise or anti-clockwise and its terminal side coincides, a coterminal angle is generated.

The initial side of the ray is known as the initial side, while the terminal side is known as the terminal side. When the difference between two angles is a multiple of 360 degrees, the two angles are said to be coterminal. The procedures outlined below can be used to calculate both the positive and negative coterminal angles of a given angle.

  1. Positive Coterminal Angles

To the given angle, multiply it by 360 degrees. The positive coterminal angles of will be + 360, + 720, etc.

If the angle is supplied in radians, we multiply it by two to get the positive coterminal angles.

  1. Negative Coterminal Angles

Subtract 360-degree multiples from the provided angle. The negative coterminal angles of will be – 360, – 720, and so on.

We remove multiples of 2 from to determine the negative coterminal angles when is represented in radians.

Coterminal Angle Calculator Solved Example

Example 1: Find coterminal angles for 60° and verify them using the coterminal angles calculator.


Given angle = 60°

Positive coterminal angles of 60° are 60° + 360°, 60° + 720°, 60° + 1080°…

= 420°, 780°, 1140°…

Negative coterminal angles of 60° are 60° – 360°, 60° – 720°, 60° – 1080°…

= -330°, -660°, -1020°…

Example 2: Find coterminal angles for -25° and verify them using the coterminal angles calculator.


Given angle = -25°

Positive coterminal angles of -25° are -25° + 360°, -25° + 720°, -25° + 1080°…

= 335°, 695°, 1055°…

Negative coterminal angles of 60° are -25° – 360°, -25° – 720°, -25° – 1080°…

= -385°, -745°, -1105° …

Similarly, you can try the coterminal angles calculator to find the coterminal angles for

Angle 45°

Angle -34°

How to find coterminal angles (Step-by-Step)?

Follow the methods below to find coterminal angles in stages:

If you’re given a radian angle (3.5 radians), you’ll need to convert it to degrees:

1 radian is 57.29 degrees, therefore 3.5*57.28=200.48.

To find an angle that is coterminal with the initial angle, multiply by 360 degrees:

200.48+360 = 560.48 degrees is a positive coterminal angle.

200.48-360 = 159.52 degrees negative coterminal angle

A Coterminal Angles Calculator, on the other hand, can be used to double-check the correctness and functionality of manual calculations.

Example 1:

Determine the coterminal angle of π/4


Given Angle: θ = π/4,

Which is in radians,

So, multiples of 2π add or subtract from it to compute its coterminal angles.

Now, subtract 2π from the angle.

= π/4 − 2π

= −7π/4

Hence, the coterminal angle of π/4 is equal to −7π/4.

Example 2:

Determine the positive and negative coterminal angle with a 495° angle.


495°−360°= 180 – 135°= 45°

A 135° angle and a 495° angle are coterminal with the 45° angle.

Coterminal Angle Calculator Tutorial

If you’re asked to discover the least positive angle coterminal with 785 degrees, you can apply the formula below. By dividing the supplied angle, 785 degrees, by 360 and rounding down to the next whole integer, the maximum number of times 360 degrees can be subtracted from 785 degrees while remaining positive is obtained. This is the number two.

Once you’ve found that number, multiply it by 360 and remove it from 785 degrees. We can use a similar calculating technique to discover the least negative angle coterminal with 526 degrees. We can do that with the exception that the dividend of the given Angle and 360 degrees must be summed up. Once this value has been discovered, it must be subtracted from the supplied angle of 526 degrees once more.

You’ll get the least negative coterminal angle this way. The same method can be used to get angles that are coterminal with radian values. If you were asked to find the least positive angle coterminal with 32 pi radian, you would follow the steps below: The dividend of the provided radian and 2pi must be rounded down to a whole number because the least positive value is being calculated.

After that, multiply this entire number by 2 pi and remove it from the given amount. As a result, the solution, we get pi. However, the calculating procedure provided below will work to obtain the least negative angle coterminal with another angle, such as 78 pi. We need the smallest negative angle in this scenario. It means that the dividends of 78 pi and 2 pi must be rounded up to the nearest whole number. For the solution to be determined, this value must be subtracted from 78 pi.

Coterminal Angle Calculator

Coterminal Angle Calculator Table


Coterminal angle of Answer
360°, 720°, -360°, -720°
361°, 721°, -359°, -719°
365°, 725°, -355°, -715°
10° 370°, 730°, -350°, -710°
15° 375°, 735°, -345°, -705°
20° 380°, 740°, -340°, -700°
25° 385°, 745°, -335°, -695°
30° (π / 6) 390°, 750°, -330°, -690
45° (π / 4) 495°, 765°, -315°, -675°
60° (π / 3 420°, 780°, -300°, -660°
75° 435°, 795°,-285°, -645°
90° (π / 2) 450°, 810°, -270°, -630°
105° 465°, 825°,-255°, -615°
120° (2π / 3) 480°, 840°, -240°, -600°
135° (3π / 4) 495°, 855°, -225°, -585°
150° (5π / 6) 510°, 870°, -210°, -570°
165° 525°, 885°, -195°, -555°
180° (π) 540°, 900°, -180°, -540°
360° (2π) 0°, 720°, -360°, -720°

Frequently Asked Questions about Coterminal Angle Calculator

How do you calculate coterminal angles?

Ans. To find the coterminal angles of a reference angle in degrees, add or subtract 360 degrees from it. If your reference angle is in pi radians, however, find the coterminal angles by adding or subtracting 2 pi from it. To learn more about how to compute coterminal angles, go to the coterminal angle calculator formula section of the page. They can also be both positive and detrimental.

How do you know if two angles are coterminal?

Ans. You can use the coterminal angle calculator for this. It will determine whether or not two angles are coterminal. The principle states that two angles are coterminal if we draw both of their terminal sides at the same location. In simple terms, this indicates that they stacked themselves on one another.

How do you find coterminal angles greater than 360 degrees?

Ans. If your angle has a value greater than 360 degrees, you need to convert it to pi radians. As a result, input the information into your coterminal angle calculator. As a result, it will give you the answer in pi radians once more. This is because dealing with such large numbers in degrees may be problematic.

Can an angle be coterminal with itself?

Ans. Yes, any angle is coterminal with itself.

How does radian measure an angle compared to the degree measure?

Ans. So, as we’ve seen, one radian equals 180 degrees. This is due to the fact that the circumference of a circle is 360 degrees, or 2 pi radians. However, you only use radians as a unit of measurement once you’ve completed high school because it requires a rudimentary understanding of trigonometry to comprehend.

What does the terminal side mean?

Ans. As a result, there is a common definition for this. It’s a straight line that we rotated around a point on another line. It generated an angle that we measured from the beginning side. In either a clockwise or counterclockwise direction, we did so.

Can the reference angle be negative?

Ans. No, because the reference angle does not exist, it cannot be negative. It has very inclusive bounds and falls between 0 and pi/2.

Are coterminal angles infinite?

Ans. Yes, it is correct. There are an unlimited number of coterminal angles for any angle. This is due to the fact that you can add or subtract 360 degrees. That’s why your coterminal angle calculator only mentions a few of them. It is virtually difficult to achieve this.

What is the smallest positive angle coterminal with?

Ans. It all relies on the perspective you’re working with. By adding or subtracting just one 2pi, you may get the smallest positive angle coterminal with any angle. So, if the angle is positive but subtracting 2pi results in a negative value, the angle equals the angle itself. If the angle is negative, however, we must look at the first positive angle after adding 2pi. So that’s how it works.

What is the importance of a reference angle?

Ans. As a result, the reference angle is critical in trigonometric computations. It’s basically a mechanism to make calculating the values of trigonometric functions at different angles easier. It is simple to calculate the value of any function at any angle using a coterminal angle calculator.