# Defining Negative Angles on the Coordinate Plane

Instructor: Melanie Olczak

Melanie has taught high school Mathematics courses for the past ten years and has a master's degree in Mathematics Education.

This lesson will explain how to define negative angles on the coordinate plane and how to draw negative angles on the coordinate plane. Examples and practice problems will be provided.

## Angles on the Coordinate Plane

Have you ever heard someone say that they've done 'a complete 180?' What does that really mean? We can only assume that this means they have turned around completely. If you are facing North and you complete a 180-degree rotation, you'd now be facing South. It doesn't matter which direction, clockwise or counterclockwise, that you turn because 180 degrees is exactly halfway from completing a full revolution.

You can think about the coordinate plane in terms of direction. We can see North, South, East and West.

We can turn in two different directions, clockwise or counterclockwise. Angles that turn in a counterclockwise direction are positive angles, while angles that turn in a clockwise direction are negative.

All angles have an initial side and a terminal side. The initial side of the angle is where the angle starts. Angles on the coordinate plane have an initial side on the positive x-axis. If we think of it in terms of directions, all angles start on East.

The terminal side of the angle is the side where the angle ends. The terminal side of the angle can be in any quadrant of the coordinate plane. There are four quadrants on the coordinate plane, and they are labeled in a counterclockwise direction.

Positive angles on the coordinate plane are angles that go in a counterclockwise direction. Notice that the initial side of the angle is on the positive x-axis and the terminal side is in the third quadrant. The arrow is pointing in a counterclockwise direction, so this angle is positive.

Negative angles on the coordinate plane are angles that go in a clockwise direction. Notice the initial side of the angle is on the positive x-axis and the terminal side is in the third quadrant like the positive angle in the previous example. However the arrow is pointing in the other direction, clockwise, so this angle would be negative. There are always at least two angles that have the same initial and terminal side called coterminal angles.

## Angle Measures

Some common angle measures correspond to the axes on the coordinate plane. One full revolution, or circle, measures 360 degrees. If we only turn halfway, we are making an angle of 180 degrees. If we go only one quarter, we make an angle of 90 degrees.

Likewise, if we complete these turns in a clockwise direction, the angles become negative.

We can create any angle between these common angles.

## Examples:

1. Draw -45 degrees on the coordinate plane.

Since the angle is negative, we know we are turning in a clockwise direction. We also know that our initial side is on the positive x-axis. Where would 45 degrees be?

Well, it is halfway to 90 degrees, so this angle will have a terminal side in the fourth quadrant.

2. Draw -150 degrees on the coordinate plane.

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