How to Calculate Coterminal Angle

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Coterminal angles are a core topic when studying trigonometry. The coterminal angle formula helps us find all coterminal angles and find infinite angle solutions for a trigonometric problem or equation.

Coterminal angles are those that share the initial and terminal side. This means they are in the same position in the unit circle:

In order to talk about this topic, it is important to note that angles can be written in two different ways:

  1. Degrees  for example: 90°; 180°, 360°, etc.
  2. Radians  for example: \frac{\pi }{2}; 2\pi, etc.

To understand the relationship between radians and degrees, and how to convert degrees to radians and radians to degrees, the following should be beard in mind: π=180°.

Having introduced radians, we will now focus on coterminal angles. This article will show you the coterminal angle formula, how to calculate coterminal angles and it will give you examples for you to follow step by step.

Coterminal Angle Formula

Angle X is coterminal of angle Y if X = Y + 2π k, k being a constant.

This means that if we add k times 2π to an angle, you will get k solutions to a trigonometric function. 

In this way, trigonometric equations and/or problems will have infinite solutions. To find them, we will need to use the coterminal angle formula. 

Coterminal Angle Examples

The question you may be asking yourself is: perfect, but, how do we apply this to a real trigonometric problem?

We will proceed to see an examples together:

Sin x = ½

By the sine, cosine and tangent unit circle table (shown below), we know that, in this case, for 0° to 90° (π/2), x=30° (π/6 in radians)

Using our unit circle knowledge, we can also find the second solution for sin x = ½ . To do so, we need to use the knowledge that, if x is in the first quadrant, it will share sine with an angle in the second quadrant. 

To find this angle, we need to do 180° (π) – angle (30°or π/6).

This gives us our second solution:

X = 150° or (5π/6)

Now, here is where the coterminal formula comes in. 

X = 30° and 150° are solutions for the problem within the limit of ONE unit circle (0° to 360°). What happens if we consider 0° to 720° or 0° to 1080° or infinite degrees for the scope of our answer?

In this case, we will use the coterminal formula for EACH angle. 

X = Y + 2π k

K will depend on the amount of unit circles we consider. Let’s say we consider 0° to 720° s the scope of our answer, in this case, our solutions would be:

X = 30°

X = 30° + 360° (2π)

X = 150°

X = 150° + 360° (2π)

Normally, we will find that problems require infinite solutions, this means, they have in mind all the possible angles.

In this case, solution for this problem would be:

X = 30° + 360° (k)

X = 150° + 360° (k)

With k belonging to the natural numbers from 0 to infinity. 

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