Coterminal Angles

In this lesson, we will look at angles at standard position and coterminal angles.

An angle is said to be in standard position if it is drawn on the Cartesian plane (x-y plane) on the positive x-axis and turning counter-clockwise (anti-clockwise).

The initial side of an angle is the ray where the measurement of an angle starts.

The terminal side of an angle is the ray where the measurement of an angle ends.

Co-terminal angles are angles which when drawn at standard position share a terminal side.  For example, 30°, -330°, 390° are all coterminal.

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 x 360° × k  if θ is given in degrees
Coterminal of θ = θ x 2πi ×k if θ is given in radians

Two angles are coterminal if the difference between them is a multiple of 360° or 2π.

Example:

Determine if the following pairs of angles are coterminal

a) 10°, 370°
b) –520°, 200°
c) –600°, –60°

Solution:

a) 10° – 370° = –360° = –1(360°), which is a multiple of 360°
So, 10° and 370° are coterminal

b) –520° – 200° = –720° = –2(360°), which is a multiple of 360°
So, –520 and 200° are coterminal

c) –600° – (–60°) = –540°, which is not a multiple of 360°
So, –600° and –60° are not coterminal

Videos

Basic Cartesian Coordinate System, Angles in Standard Position, Coterminal Angles

Coterminal Angles, Complementary Angles, Supplementary Angles in Radians

Finding coterminal angles -
Professor Edward Burger explains finding coterminal angles

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