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Conversion between Degrees and Radians

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Introduction to the Unit Circle and Radian Measure
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An odd-shaped house in Asia is built at a 135^\circ angle. How many radians is this angle equal to?

Guidance

The unit circle is the circle centered at the origin with radius equal to one unit. This means that the distance from the origin to any point on the circle is equal to one unit.

Using the unit circle, we can define another unit of measure for angles, radians. Radian measure is based upon the circumference of the unit circle. The circumference of the unit circle is 2 \pi ( 2 \pi r , where r=1 ). So a full revolution, or 360^\circ , is equal to 2 \pi radians. Half a rotation, or 180^\circ is equal to \pi radians.

One radian is equal to the measure of \theta , the rotation required for the arc length intercepted by the angle to be equal to the radius of the circle. In other words the arc length is 1 unit for \theta=1 radian.

We can use the equality, \pi=180^\circ to convert from degrees to radians and vice versa.

To convert from degrees to radians, multiply by \frac{\pi}{180^\circ} .

To convert from radians to degrees, multiply by \frac{180^\circ}{\pi} .

Example A

a. Convert 250^\circ to radians.

b. Convert 3 \pi to degrees.

Solution:

a. To convert from degrees to radians, multiply by \frac{\pi}{180^\circ} . So, \frac{250 \pi}{180}=\frac{25 \pi}{18} .

b. To convert from radians to degrees, multiply by \frac{180^\circ}{\pi} . So, 3 \pi \times \frac{180^\circ}{\pi}=3 \times 180^\circ=540^\circ .

Example B

Find two angles, one positive and one negative, coterminal to \frac{5 \pi}{3} and find its reference angle, in radians.

Solution: Since we are working in radians now we will add/subtract multiple of 2 \pi instead of 360^\circ . Before we can add, we must get a common denominator of 3 as shown below.

\frac{5 \pi}{3} + 2 \pi = \frac{5 \pi}{3} + \frac{6 \pi}{3} = \frac{11 \pi}{3} \quad and \quad \frac{5 \pi}{3} - 2 \pi = \frac{5 \pi}{3} - \frac{6 \pi}{3} =- \frac{\pi}{3}

Now, to find the reference angle, first determine in which quadrant \frac{5 \pi}{3} lies. If we think of the measures of the angles on the axes in terms of \pi and more specifically, in terms of \frac{\pi}{3} , this task becomes a little easier.

Consider \pi is equal to \frac{3 \pi}{3} and 2 \pi is equal to \frac{6 \pi}{3} as shown in the diagram. Now we can see that the terminal side of \frac{5 \pi}{3} lies in the fourth quadrant and thus the reference angle will be:

\frac{6 \pi}{3} - \frac{5 \pi}{3} = \frac{\pi}{3}

Example C

Find two angles coterminal to \frac{7 \pi}{6} , one positive and one negative, and find its reference angle, in radians.

Solution: This time we will add multiples of 2 \pi with a common denominator of 6, or \frac{2 \pi}{1} \times \frac{6}{6} = \frac{12 \pi}{6} . For the positive angle, we add to get \frac{7 \pi}{6} + \frac{12 \pi}{6} = \frac{19 \pi}{6} . For the negative angle, we subtract to get \frac{7 \pi}{6} - \frac{12 \pi}{6} = \frac{5 \pi}{6} .

In this case \pi is equal to \frac{6 \pi}{6} and 2 \pi is equal to \frac{12 \pi}{6} as shown in the diagram. Now we can see that the terminal side of \frac{7 \pi}{6} lies in the third quadrant and thus the reference angle will be:

\frac{7 \pi}{6} - \frac{6 \pi}{6} = \frac{\pi}{6}

Concept Problem Revisit

To convert from degrees to radians, multiply by \frac{\pi}{180^\circ} . So, \frac{135 \pi}{180}=\frac{3 \pi}{4} .

Guided Practice

1. Convert the following angle measures from degrees to radians.

a. -45^\circ

b. 120^\circ

c. 330^\circ

2. Convert the following angle measures from radians to degrees.

a. \frac{5 \pi}{6}

b. \frac{13 \pi}{4}

c. -\frac{5 \pi}{2}

3. Find two coterminal angles to \frac{11 \pi}{4} , one positive and one negative, and its reference angle.

Answers

1. a. -45^\circ \times \frac{\pi}{180^\circ}=-\frac{\pi}{4}

b. 120^\circ \times \frac{\pi}{180^\circ}=\frac{2 \pi}{3}

c. 330^\circ \times \frac{\pi}{180^\circ}=\frac{11 \pi}{6}

2. a. \frac{5 \pi}{6} \times \frac{180^\circ}{\pi}=150^\circ

b. \frac{13 \pi}{4} \times \frac{180^\circ}{\pi}=585^\circ

c. -\frac{5 \pi}{2} \times \frac{180^\circ}{\pi}=-450^\circ

3. There are many possible coterminal angles, here are some possibilities:

positive coterminal angle: \frac{11 \pi}{4} + \frac{8 \pi}{4} = \frac{19 \pi}{4} or \frac{11 \pi}{4} - \frac{8 \pi}{4} = \frac{3 \pi}{4} ,

negative coterminal angle: \frac{11 \pi}{4} - \frac{16 \pi}{4} = -\frac{5 \pi}{4} or \frac{11 \pi}{4} - \frac{24 \pi}{4} = -\frac{13 \pi}{4}

Using the coterminal angle, \frac{3 \pi}{4} , which is \frac{\pi}{4} from \frac{4 \pi}{4} . So the terminal side lies in the second quadrant and the reference angle is \frac{\pi}{4} .

Vocabulary

Radian
When the arc length formed by a central angle is equal in length to the radius of the circle.

Practice

For problems 1-5, convert the angle from degrees to radians. Leave answers in terms of \pi .

  1. 135^\circ
  2. 240^\circ
  3. -330^\circ
  4. 450^\circ
  5. -315^\circ

For problems 6-10, convert the angle measure from radians to degrees.

  1. \frac{7 \pi}{3}
  2. -\frac{13 \pi}{6}
  3. \frac{9 \pi}{2}
  4. -\frac{3 \pi}{4}
  5. \frac{5 \pi}{6}

For problems 11-15, find two coterminal angles (one positive, one negative) and the reference angle for each angle in radians.

  1. \frac{8 \pi}{3}
  2. \frac{11 \pi}{4}
  3. -\frac{\pi}{6}
  4. \frac{4 \pi}{3}
  5. -\frac{17 \pi}{6}

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