Lesson Objectives

- Learn about radian measure
- Learn how to convert between degrees and radians

## Radian Measure

In this lesson, we will learn about radian measure and show how to convert between degrees and radians. So far in our course, we have measured our angles in degrees. It is often useful to measure angles using radians instead of degrees. This will allow us to treat trigonometric functions as functions with domains of real numbers.

Let's now apply this concept to an angle in standard position: Let's now extend our concept into the realm of 2 radians. What would an angle of measure of 2 radians look like? This angle would now intercept an arc that is equal to twice the radius of the circle: In our image above, θ = 2 radians.

In general, if our angle θ is a central angle of a circle of radius r, and θ intercepts an arc of length s, then we can obtain our radian measure of θ as: $$θ=\frac{s}{r}\text{radians}$$ In the case where the arc length s is equal to the radius, we obtain an angle where θ is exactly 1 radian. $$θ=\frac{r}{r}\text{radians}=1 \hspace{.1em}\text{radian}$$

Divide by 180: $$1°=\frac{π}{180}\hspace{.1em}\text{radians}$$ Divide by $π$: $$1 \hspace{.1em}\text{radian}=\frac{180°}{π}$$

Example #1: Convert each degree measure into radians. $$150°$$ $$150°=150\left(\frac{π}{180}\text{radian}\right)$$ $$\require{cancel}150°=5\cancel{150}\left(\frac{π}{6\cancel{180}}\text{radian}\right)$$ $$150°=\frac{5π}{6}\text{radians}$$ Example #2: Convert each degree measure into radians. $$260°$$ $$260°=260\left(\frac{π}{180}\text{radian}\right)$$ $$260°=13\cancel{260}\left(\frac{π}{9\cancel{180}}\text{radian}\right)$$ $$260°=\frac{13π}{9}\text{radians}$$

Example #3: Convert each radian measure into degrees. $$-\frac{157π}{36}\hspace{.1em}\text{radians}$$

### What is a Radian?

The radian is based on the radius of a circle. Recall that the distance from the center of a circle to any point on the circle is known as the radius. To understand where radians come from, let's begin by sketching a circle with a radius r. Next, we will make a copy of our radius and extend this vertically: Lastly, let's bend our copy of the radius so that it lies on the circle: An angle with its vertex at the center of a circle that intercepts an arc on the circle equal in length to the radius of the circle has a measure of exactly 1 radian: In our image above, θ = 1 radian.Let's now apply this concept to an angle in standard position: Let's now extend our concept into the realm of 2 radians. What would an angle of measure of 2 radians look like? This angle would now intercept an arc that is equal to twice the radius of the circle: In our image above, θ = 2 radians.

In general, if our angle θ is a central angle of a circle of radius r, and θ intercepts an arc of length s, then we can obtain our radian measure of θ as: $$θ=\frac{s}{r}\text{radians}$$ In the case where the arc length s is equal to the radius, we obtain an angle where θ is exactly 1 radian. $$θ=\frac{r}{r}\text{radians}=1 \hspace{.1em}\text{radian}$$

## How to Convert Between Degrees and Radians

We know that an angle of 360° corresponds to one complete rotation and also one complete circle: We know the formula for the circumference of a circle or the distance around the circle is given by: $$C=2πr$$ Therefore, an angle of 360°, which is one complete circle, intercepts an arc equal in length to 2$π$ times the radius of the circle: $$θ=\frac{2πr}{r}\text{radians}=2π\hspace{.1em}\text{radians}$$ So an angle of measure of 360° is exactly equal to 2$π$ radians. Let's suppose we had a straight angle or an angle of measure of 180°. This angle measure is 1/2 of 360°: $$180°=\frac{1}{2}\cdot 2π \hspace{.1em}\text{radians}=π \hspace{.1em}\text{radians}$$ $$180°=π \hspace{.1em}\text{radians}$$ We can use the above relationship to develop a method for converting between degrees and radians:Divide by 180: $$1°=\frac{π}{180}\hspace{.1em}\text{radians}$$ Divide by $π$: $$1 \hspace{.1em}\text{radian}=\frac{180°}{π}$$

### Convert Degrees to Radians

To convert from degrees to radians, we multiply a degree measure by: $$\frac{π}{180}\text{radian}$$ Let's look at a few examples.Example #1: Convert each degree measure into radians. $$150°$$ $$150°=150\left(\frac{π}{180}\text{radian}\right)$$ $$\require{cancel}150°=5\cancel{150}\left(\frac{π}{6\cancel{180}}\text{radian}\right)$$ $$150°=\frac{5π}{6}\text{radians}$$ Example #2: Convert each degree measure into radians. $$260°$$ $$260°=260\left(\frac{π}{180}\text{radian}\right)$$ $$260°=13\cancel{260}\left(\frac{π}{9\cancel{180}}\text{radian}\right)$$ $$260°=\frac{13π}{9}\text{radians}$$

### Convert Radians to Degrees

To convert from radians to degrees, we multiply a radian measure by: $$\frac{180°}{π}$$ Let's look at a few examples.Example #3: Convert each radian measure into degrees. $$-\frac{157π}{36}\hspace{.1em}\text{radians}$$

$$-\frac{157π}{36}\hspace{.1em}\text{radians}=-\frac{157π}{36}\left(\frac{180°}{π}\right)=-\frac{157\cancel{π}}{\cancel{36}}\left(\frac{5\cancel{180}°}{\cancel{π}}\right)=-785°$$

Example #4: Convert each radian measure into degrees. $$\frac{43π}{12}\hspace{.1em}\text{radians}$$ $$\frac{43π}{12}\hspace{.1em}\text{radians}=\frac{43π}{12}\left(\frac{180°}{π}\right)=\frac{43\cancel{π}}{\cancel{12}}\left(\frac{15\cancel{180}°}{\cancel{π}}\right)=645°$$

### Common Degree to Radian Conversions

It is helpful to memorize some of the most commonly used angle measures in degrees converted to radians. In our image below, the inner set of numbers are in degrees and the outer set will be the corresponding radian measure.#### Skills Check:

Example #1

Convert each radian measure into degrees. $$\frac{7π}{12}$$

Please choose the best answer.

A

120°

B

105°

C

115°

D

127°

E

215°

Example #2

Convert each radian measure into degrees. $$\frac{25π}{18}$$

Please choose the best answer.

A

250°

B

500°

C

245°

D

240°

E

115°

Example #3

Convert each degree measure into radians. $$570°$$

Please choose the best answer.

A

$$\frac{25π}{9}$$

B

$$\frac{14π}{3}$$

C

$$\frac{35π}{12}$$

D

$$\frac{32π}{9}$$

E

$$\frac{19π}{6}$$

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