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Converting degrees to radians

Convert 15 degrees to radians.

In this example, we start with degrees and want radians, so we again set up a proportion, but we substitute the given information into a different part of the proportion.

θ 180 = θ R π 15 180 = θ R π 15 π 180 = θ R π 12 = θ R
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Convert 126° to radians.

7 π 10

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Finding coterminal angles

Converting between degrees and radians can make working with angles easier in some applications. For other applications, we may need another type of conversion. Negative angles and angles greater than a full revolution are more awkward to work with than those in the range of to 360° , or 0 to 2 π . It would be convenient to replace those out-of-range angles with a corresponding angle within the range of a single revolution.

It is possible for more than one angle to have the same terminal side. Look at [link] . The angle of 140° is a positive angle, measured counterclockwise. The angle of –220° is a negative angle, measured clockwise. But both angles have the same terminal side. If two angles in standard position have the same terminal side, they are coterminal angles    . Every angle greater than 360° or less than is coterminal with an angle between and 360° , and it is often more convenient to find the coterminal angle within the range of to 360° than to work with an angle that is outside that range.

A graph showing the equivalence between a 140 degree angle and a negative 220 degree angle.  The 140 degrees angle is a counterclockwise rotation where the 220 degree angle is a clockwise rotation.
An angle of 140° and an angle of –220° are coterminal angles.

Any angle has infinitely many coterminal angles because each time we add 360° to that angle—or subtract 360° from it—the resulting value has a terminal side in the same location. For example, 100° and 460° are coterminal for this reason, as is −260° .

An angle’s reference angle is the measure of the smallest, positive, acute angle t formed by the terminal side of the angle t and the horizontal axis. Thus positive reference angles have terminal sides that lie in the first quadrant and can be used as models for angles in other quadrants. See [link] for examples of reference angles for angles in different quadrants.

Four side-by-side graphs. First graph shows an angle of t in quadrant 1 in its normal position. Second graph shows an angle of t in quadrant 2 due to a rotation of pi minus t. Third graph shows an angle of t in quadrant 3 due to a rotation of t minus pi. Fourth graph shows an angle of t in quadrant 4 due to a rotation of two pi minus t.

Coterminal and reference angles

Coterminal angles are two angles in standard position that have the same terminal side.

An angle’s reference angle    is the size of the smallest acute angle, t , formed by the terminal side of the angle t and the horizontal axis.

Given an angle greater than 360° , find a coterminal angle between and 360°

  1. Subtract 360° from the given angle.
  2. If the result is still greater than 360° , subtract 360° again till the result is between and 360° .
  3. The resulting angle is coterminal with the original angle.

Finding an angle coterminal with an angle of measure greater than 360°

Find the least positive angle θ that is coterminal with an angle measuring 800° , where θ < 360 ° .

An angle with measure 800° is coterminal with an angle with measure 800 360 = 440° , but 440° is still greater than 360° , so we subtract 360° again to find another coterminal angle: 440 360 = 80° .

The angle θ = 80° is coterminal with 800° . To put it another way, 800° equals 80° plus two full rotations, as shown in [link] .

A graph showing the equivalence between an 80-degree angle and an 800-degree angle where the 800 degree angle is two full rotations and has the same terminal side position as the 80 degree.
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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