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5.1 Angles  (Page 2/29)

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Drawing an angle in standard position always starts the same way—draw the initial side along the positive x -axis. To place the terminal side of the angle, we must calculate the fraction of a full rotation the angle represents. We do that by dividing the angle measure in degrees by 360°. For example, to draw a 90° angle, we calculate that   90° 360° = 1 4 .   So, the terminal side will be one-fourth of the way around the circle, moving counterclockwise from the positive x -axis. To draw a 360° angle, we calculate that   360° 360° = 1.   So the terminal side will be 1 complete rotation around the circle, moving counterclockwise from the positive x -axis. In this case, the initial side and the terminal side overlap. See [link] .

Side by side graphs. Graph on the left is a 90 degree angle and graph on the right is a 360 degree angle. Terminal side and initial side are labeled for both graphs.

Since we define an angle in standard position    by its initial side, we have a special type of angle whose terminal side lies on an axis, a quadrantal angle . This type of angle can have a measure of 0°, 90°, 180°, 270° or 360°. See [link] .

Four side by side graphs. First graph shows angle of 0 degrees. Second graph shows an angle of 90 degrees. Third graph shows an angle of 180 degrees. Fourth graph shows an angle of 270 degrees.
Quadrantal angles are angles in standard position whose terminal side lies along an axis. Examples are shown.

Quadrantal angles

Quadrantal angles are angels in standard position whose terminal side lies on an axis, including 0°, 90°, 180°, 270°, or 360°.

Given an angle measure in degrees, draw the angle in standard position.

  1. Express the angle measure as a fraction of 360°.
  2. Reduce the fraction to simplest form.
  3. Draw an angle that contains that same fraction of the circle, beginning on the positive x -axis and moving counterclockwise for positive angles and clockwise for negative angles.

Drawing an angle in standard position measured in degrees

  1. Sketch an angle of 30° in standard position.
  2. Sketch an angle of −135° in standard position.
  1. Divide the angle measure by 360°.
    30° 360° = 1 12

    To rewrite the fraction in a more familiar fraction, we can recognize that

    1 12 = 1 3 ( 1 4 )

    One-twelfth equals one-third of a quarter, so by dividing a quarter rotation into thirds, we can sketch a line at 30° as in [link] .

    Graph of a 30 degree angle.
  2. Divide the angle measure by 360°.
    − 135° 360° = − 3 8

    In this case, we can recognize that

    − 3 8 = − 3 2 ( 1 4 )

    Negative three-eighths is one and one-half times a quarter, so we place a line by moving clockwise one full quarter and one-half of another quarter, as in [link] .

    Graph of a negative 135 degree angle.
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Show an angle of 240° on a circle in standard position.

Graph of a 240 degree angle.
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Converting between degrees and radians

Dividing a circle into 360 parts is an arbitrary choice, although it creates the familiar degree measurement. We may choose other ways to divide a circle. To find another unit, think of the process of drawing a circle. Imagine that you stop before the circle is completed. The portion that you drew is referred to as an arc. An arc may be a portion of a full circle, a full circle, or more than a full circle, represented by more than one full rotation. The length of the arc around an entire circle is called the circumference of that circle.

The circumference of a circle is   C = 2 π r .   If we divide both sides of this equation by   r , we create the ratio of the circumference to the radius, which is always   2 π   regardless of the length of the radius. So the circumference of any circle is   2 π ≈ 6.28   times the length of the radius. That means that if we took a string as long as the radius and used it to measure consecutive lengths around the circumference, there would be room for six full string-lengths and a little more than a quarter of a seventh, as shown in [link] .

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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