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Conversion between Degrees and Radians

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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 75^\circ 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.

Watch This

James Sousa Example: Converting Angles in Degree Measure to Radian Measure

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: \pi \ \text{radians} = 180^\circ . If you divide both sides of this equation by \pi , you will have the conversion formula:

\text{radians} \times \frac{180}{\pi}=\text{degrees}

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

\text{degrees} \times \frac{\pi}{180}=\text{radians}

Example A

Convert \frac{11\pi}{3} to degree measure.

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

Here is what it would look like using the formula:

\text{radians} \times \frac{180}{\pi} = \text{degrees}

Example B

Convert -120^\circ to radian measure. Leave the answer in terms of \pi .

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

and reducing to lowest terms gives us -\frac{2\pi}{3}

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

Example C

Express \frac{11\pi}{12} radians terms of degrees.

\text{radians} \times \frac{180}{\pi}=\text{degrees}

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

Vocabulary

Radian: A radian (abbreviated rad) is the angle created by bending the radius length around the arc of a circle.

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 \pi .

240^\circ , 270^\circ , 315^\circ , -210^\circ , 120^\circ

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

15^\circ , -450^\circ , 72^\circ , 720^\circ , 330^\circ

3. Convert the following radian measures to degrees

\frac{\pi}{2} , \frac{11\pi}{5} , \frac{2\pi}{3} , 5\pi , \frac{7\pi}{2}

Solutions:

1. \frac{4\pi}{3} , \frac{3\pi}{2} , \frac{7\pi}{4} , -\frac{7\pi}{6} , \frac{2\pi}{3}

2. \frac{\pi}{12} , -\frac{5\pi}{2} , \frac{\pi}{5} , 4 \pi , \frac{11\pi}{6}

3. 90^\circ , 396^\circ , 120^\circ , 540^\circ , 630^\circ

Concept Problem Solution

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

\text{degrees} \times \frac{\pi}{180}=\text{radians}

you can find the solution to convert 75^\circ to radians:

75^\circ \times \frac{\pi}{180}= \frac{75\pi}{180} = \frac{5\pi}{12}

Practice

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

  1. 90^\circ
  2. 360^\circ
  3. 50^\circ
  4. 110^\circ
  5. 495^\circ
  6. -85^\circ
  7. -120^\circ

Convert the following radian measures to degrees.

  1. \frac{5\pi}{12}
  2. \frac{3\pi}{5}
  3. \frac{8\pi}{15}
  4. \frac{7\pi}{10}
  5. \frac{5\pi}{2}
  6. 3\pi
  7. \frac{7\pi}{2}
  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?

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