How Co-terminal Works

For an angle to be in a standard position, it needs to rotate counterclockwise on the positive x-axis of a Cartesian plane (x and y-axis). The initial sides of an angle in a standard position are the line where the magnitude of an angle begins. On the other hand, the terminal sides of an angle in a standard position are the line where the magnitude of an angle stops.

In other words, co-terminal angles are angles that have the same initial and terminal sides when placed in a standard position. Examples of such co-terminal angles are 30°, -330°, and 390°. You can find the co-terminal angles of a specific angle by applying the formula below. To find the co-terminal angle of a specific angle represented by θ, you have to add or subtract it from 360° or 2π radians.

That is to say, co-terminal of θ = θ + 360° × k, if the value of θ is written in degrees. However, co-terminal of θ = θ + 2π × k if the value of θ is written in radians. Two distinct angles are said to be co-terminal angles if the difference between them is a sum of 360° or 2π radians.

Example of Co-terminal

If the co-terminal angle of two angles, for example, 40° and 400° must be determined then, 40° subtracts from 400°, which results in -360°. That is to say that 40° and 400° are co-terminal angles. However, if this same method is applied, you will find out that some angles are not co-terminal angles, -600° and -60° for example.

Types of Co-terminal Angles

The use of co-terminal angles is very common in mathematical calculations that involve trigonometry. In trigonometry, co-terminal angles help to find the following.

  1. Degrees: If you have a given angle, you can use the co-terminal angles to find its value in degrees. In this case, the given angle must be subtracted or added to 360°.
  2. Radians: Co-terminal angles form an angle of radian 2π. Just like the way it is with degrees, you can find the radian of a given angle by subtracting from or adding it to 2π.

Significance of Co-terminal Angles

Trigonometry uses co-terminal angles to find the degree or radian value. That is because it has the same value for sine, cosine, and tangent. You must know that trigonometry uses the functions of the sin, cos, and tan angles frequently. It means that using co-terminal angles is an easier alternative during calculations involving trigonometry.

Since co-terminal angles have the identical value for sin, cos, and tan angles, the co-terminal angle value can be easily found and substituted into the trigonometric calculation. This method is efficient and accurate and saves a lot of time, which is very important in mathematical calculations of any kind. But that's not all. In trigonometry, co-terminal angles help to find the degree or radian of any given angle.