Sketching Angles and Radians

Sketching Angles and Radians

Goal: To sketch/identify negative and positive angles, and to convert angles from degrees to radians and vice versa.

Sketching Angles

Have you ever heard someone say that they've done 'a complete 180?' 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 you turn because 180 degrees is exactly halfway from completing a full revolution. You can think of the coordinate plane in terms of North, South East, and West.

Sketching Angles

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

Sketching Angles

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.

Sketching Angles

The terminal side of the angle is the side where the angle ends, or terminates. 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.

Sketching Angles

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.

Sketching Angles

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.

Sketching Angles

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.

Example 1

• Is this a positive or a negative angle?

• Positive since the arrow goes in the counterclockwise direction

• What is the measure of this angle?

• 35 + 90 = 125 degress

Sketching Angles

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

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.

Draw -150 degrees on the coordinate plane.

• Since this angle is negative, we know that we are turning in a clockwise direction. Our initial side never changes, so we are starting on the positive x-axis. Between what two common angles is 150? Which one is it closer to?

• 150 is between 90 and 180, closer to 180. So we can conclude that the terminal side will be in the third quadrant but closer to 180 degrees.

Draw -300 degrees on the coordinate plane.

• This angle is negative, so again we know we are turning in a clockwise direction, with our initial side on the positive x-axis.

• 300 degrees is between 270 and 360. 300 degrees is close to 360 degrees but closer to 270, so the terminal side will be in the first quadrant closer to -270.

Summary

• Angles can be drawn on the coordinate plane. All angles have an initial side, which is where the angle starts, on the positive x-axis. The terminal side of any angle is where the angle ends, and that can be anywhere on the coordinate plane. Angles on the coordinate plane are either positive or negative.

• Positive angles turn in a counterclockwise direction.

• Negative angles turn in a clockwise direction.

• We can draw any angle on the coordinate plane by determining which direction it goes and how close it is to the common angle measures that we already know: 90, 180, 270 and 360 degrees.

Radians

A radian is the standard unit of angular measure. This is because they are easiest to work with mathematically. Because the radian is related to the radius of a circle, radian measures can be used in trigonometric calculations easier than degrees.

Radians

Recall that the arc length of a complete rotation is the circumference, where the formula is equal to 2π times the length of the radius. So, in terms of radian measure, a complete rotation (360 degrees) is 2π radians.

Since 2π radians = 360°, we can create a conversion factor.

Radians

Convert 125° to radians.

Multiple Choice

Convert 150° to radians.

• $\frac{5\pi}{12}$ radians

• $30\pi$ radians

• $330\pi$ radians

• $\frac{5\pi}{6}$ radians