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Conversion between Degrees and Radians

Convert between radians and degrees

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Conversion between Degrees and Radians

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?

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: \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*}

 

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

1. 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*}

2. 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*}.

3. 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.

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

\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*}

 

Example 2

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*}

\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*}

Example 3

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*}

 

\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*}

 

Example 4

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*}

\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*}

Review

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?

Review (Answers)

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

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Vocabulary

Degree

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

radian

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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