IT - Degrees and Radians Lesson

Degrees and Radians

In the last lesson, we learned that there is a beginning and an end to an angle. In this module, we are going to consider all angles to be in standard position which means the vertex is at the origin and the initial side of the angle lies on the positive x-axis. Later in the course, when solving navigation problems, we will consider angles measured from the positive y-axis (due north).

Radian: A radian is the measure of the central angle of a circle subtended by an arc of equal length to the radius. Watch the video to see how to mark off three radians on a circle.

 

How many radians are in a circle? In any circle, there are approximately 6.28 radians or, more specifically, $2\pi$ radians.

What is the relationship between degrees and radians?

**SPECIAL NOTE** In the video below, we are asked to find AG and it should say, find the length of arc AG.

Convert from degrees to radians: Multiply by $\frac{\pi}{180}$

Convert from radians to degrees: Multiply by $\frac{180}{\pi}$

Example

Given angle $\theta=135^\circ$, we can convert it to radians by multiplying by $\frac{\pi}{180}$.

$135\left(\frac{\pi}{180}\right)=\frac{135\pi}{180}=\frac{3\pi}{4}$

Given angle $\theta=\frac{\pi}{2}$, we can convert it to degrees by multiplying by $\frac{180}{\pi}$.

$\frac{\pi}{2}\left(\frac{180}{\pi}\right)=\frac{180\pi}{2\pi}=90^\circ$

Important Fact: If an angle is in degrees, there will be a degree symbol. If an angle is in radians there will be no symbol!

Let's check your understanding. Match the angles below:

In the last lesson, we learned about co-terminal angles and know that you can find the co-terminal angle of an angle in degrees by adding or subtracting 360°. If an angle is given to you in radians, you can add or subtract  to find a coterminal angle.

A reference angle is the positive acute angle formed by the terminal side of an angle and the x-axis.

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