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

### Conversion between Degrees and Radians

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:

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

Let's look at a few problems where we convert between degrees and radians.

1. Convert

From the last section, you should recognize that this angle is a multiple of

Here is what it would look like using the formula:

2. Convert

and reducing to lowest terms gives us

You could also have noticed that 120 is

3. Express

Note: Sometimes students have trouble remembering if it is

### Examples

#### Example 1

Earlier, you were asked is there a way to translate the instructions in degrees to radians.

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

you can find the solution to convert

#### Example 2

Convert the following degree measures to radians. All answers should be in terms of

#### Example 3

Convert the following degree measures to radians. All answers should be in terms of

#### Example 4

Convert the following radian measures to degrees

\begin{align*}\frac{\pi}{2}\end{align*}

\begin{align*}90^\circ\end{align*}

### Review

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

- \begin{align*}90^\circ\end{align*}
90∘ - \begin{align*}360^\circ\end{align*}
- \begin{align*}50^\circ\end{align*}
- \begin{align*}110^\circ\end{align*}
- \begin{align*}495^\circ\end{align*}
- \begin{align*}-85^\circ\end{align*}
- \begin{align*}-120^\circ\end{align*}

Convert the following radian measures to degrees.

- \begin{align*}\frac{5\pi}{12}\end{align*}
- \begin{align*}\frac{3\pi}{5}\end{align*}
- \begin{align*}\frac{8\pi}{15}\end{align*}
- \begin{align*}\frac{7\pi}{10}\end{align*}
- \begin{align*}\frac{5\pi}{2}\end{align*}
- \begin{align*}3\pi\end{align*}
- \begin{align*}\frac{7\pi}{2}\end{align*}
- 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?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.2.