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# 2.2: Conversion between Degrees and Radians

Difficulty Level: At Grade Created by: CK-12
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Practice Conversion between Degrees and Radians
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You are hard at work in the school science lab when your teacher asks you to turn a knob on a detector you are using \begin{align*}75^\circ\end{align*} degrees. Unfortunately, you have been working in radians for a while, and so you're having trouble remembering how far to turn the knob. Is there a way to translate the instructions in degrees to radians?

Read this Concept, and at the conclusion you'll be able to accomplish this task and turn the knob the appropriate amount.

### Guidance

Since degrees and radians are different ways of measuring the distance moved around the circumference of a circle, it is reasonable to suppose that there is a conversion formula between these two units. This formula works for all degrees and radians. Remember that: \begin{align*}\pi \ \text{radians} = 180^\circ\end{align*}. If you divide both sides of this equation by \begin{align*}\pi\end{align*}, you will have the conversion formula:

\begin{align*}\text{radians} \times \frac{180}{\pi}=\text{degrees}\end{align*}

If we have a degree measure and wish to convert it to radians, then manipulating the equation above gives:

\begin{align*}\text{degrees} \times \frac{\pi}{180}=\text{radians}\end{align*}

#### Example A

Convert \begin{align*}\frac{11\pi}{3}\end{align*} to degree measure.

From the last section, you should recognize that this angle is a multiple of \begin{align*}\frac{\pi}{3}\end{align*} (or 60 degrees), so there are 11, \begin{align*}\frac{\pi}{3}\end{align*}'s in this angle, \begin{align*}\frac{\pi}{3} \times 11 = 60^\circ \times 11 = 660^\circ\end{align*}.

Here is what it would look like using the formula:

\begin{align*}\text{radians} \times \frac{180}{\pi} = \text{degrees}\end{align*}

#### Example B

Convert \begin{align*}-120^\circ\end{align*} to radian measure. Leave the answer in terms of \begin{align*}\pi\end{align*}.

\begin{align*}\text{degrees} \times \frac{\pi}{180}&=\text{radians}\\ -120^\circ \times \frac{\pi}{180}&=\frac{-120^\circ \pi}{180}\end{align*}

and reducing to lowest terms gives us \begin{align*}-\frac{2\pi}{3}\end{align*}

You could also have noticed that 120 is \begin{align*}2 \times 60\end{align*}. Since \begin{align*}60^\circ\end{align*} is \begin{align*}\frac{\pi}{3}\end{align*} radians, then 120 is 2, \begin{align*}\frac{\pi}{3}\end{align*}’s, or \begin{align*}\frac{2\pi}{3}\end{align*}. Make it negative and you have the answer, \begin{align*}-\frac{2\pi}{3}\end{align*}.

#### Example C

Express \begin{align*}\frac{11\pi}{12}\end{align*} radians terms of degrees.

\begin{align*}\text{radians} \times \frac{180}{\pi}=\text{degrees}\end{align*}

Note: Sometimes students have trouble remembering if it is \begin{align*}\frac{180}{\pi}\end{align*} or \begin{align*}\frac{\pi}{180}\end{align*}. It might be helpful to remember that radian measure is almost always expressed in terms of \begin{align*}\pi\end{align*}. If you want to convert from radians to degrees, you want the \begin{align*}\pi\end{align*} to cancel out when you multiply, so it must be in the denominator.

### Vocabulary

Degree: A degree is a unit for measuring angles in a circle. There are 360 of them in a circle.

### Guided Practice

1. Convert the following degree measures to radians. All answers should be in terms of \begin{align*}\pi\end{align*}.

\begin{align*}240^\circ\end{align*}, \begin{align*}270^\circ\end{align*}, \begin{align*}315^\circ\end{align*}, \begin{align*}-210^\circ\end{align*}, \begin{align*}120^\circ\end{align*}

2.Convert the following degree measures to radians. All answers should be in terms of \begin{align*}\pi\end{align*}.

\begin{align*}15^\circ\end{align*}, \begin{align*}-450^\circ\end{align*}, \begin{align*}72^\circ\end{align*}, \begin{align*}720^\circ\end{align*}, \begin{align*}330^\circ\end{align*}

3. Convert the following radian measures to degrees

\begin{align*}\frac{\pi}{2}\end{align*}, \begin{align*}\frac{11\pi}{5}\end{align*}, \begin{align*}\frac{2\pi}{3}\end{align*}, \begin{align*}5\pi\end{align*}, \begin{align*}\frac{7\pi}{2}\end{align*}

Solutions:

1. \begin{align*}\frac{4\pi}{3}\end{align*}, \begin{align*}\frac{3\pi}{2}\end{align*}, \begin{align*}\frac{7\pi}{4}\end{align*}, \begin{align*}-\frac{7\pi}{6}\end{align*}, \begin{align*}\frac{2\pi}{3}\end{align*}

2. \begin{align*}\frac{\pi}{12}\end{align*}, \begin{align*}-\frac{5\pi}{2}\end{align*}, \begin{align*}\frac{\pi}{5}\end{align*}, \begin{align*}4 \pi\end{align*}, \begin{align*}\frac{11\pi}{6}\end{align*}

3. \begin{align*}90^\circ\end{align*}, \begin{align*}396^\circ\end{align*}, \begin{align*}120^\circ\end{align*}, \begin{align*}540^\circ\end{align*}, \begin{align*}630^\circ\end{align*}

### Concept Problem Solution

Since you now know that the conversion for a measurement in degrees to radians is

\begin{align*}\text{degrees} \times \frac{\pi}{180}=\text{radians}\end{align*}

you can find the solution to convert \begin{align*}75^\circ\end{align*} to radians:

\begin{align*}75^\circ \times \frac{\pi}{180}= \frac{75\pi}{180} = \frac{5\pi}{12}\end{align*}

### Practice

Convert the following degree measures to radians. All answers should be in terms of \begin{align*}\pi\end{align*}.

1. \begin{align*}90^\circ\end{align*}
2. \begin{align*}360^\circ\end{align*}
3. \begin{align*}50^\circ\end{align*}
4. \begin{align*}110^\circ\end{align*}
5. \begin{align*}495^\circ\end{align*}
6. \begin{align*}-85^\circ\end{align*}
7. \begin{align*}-120^\circ\end{align*}

Convert the following radian measures to degrees.

1. \begin{align*}\frac{5\pi}{12}\end{align*}
2. \begin{align*}\frac{3\pi}{5}\end{align*}
3. \begin{align*}\frac{8\pi}{15}\end{align*}
4. \begin{align*}\frac{7\pi}{10}\end{align*}
5. \begin{align*}\frac{5\pi}{2}\end{align*}
6. \begin{align*}3\pi\end{align*}
7. \begin{align*}\frac{7\pi}{2}\end{align*}
8. Why do you think there are two different ways to measure angles? When do you think it might be more convenient to use radians than degrees?

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

Degree

A degree is a unit for measuring angles in a circle. There are 360 degrees in a circle.

A radian is a unit of angle that is equal to the angle created at the center of a circle whose arc is equal in length to the radius.

subtended arc

A subtended arc is the part of the circle in between the two rays that make the central angle.

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