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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 terminal 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 have a terminal side that lies along an axis. Examples are shown.

Quadrantal angles

An angle is a quadrantal angle    if its 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 on an xy-plane.
  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 with a clockwise rotation to the terminal side instead of counterclockwise.
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Show an angle of 240° on a circle in standard position.

Graph of a 240-degree angle with a counterclockwise rotation.
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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, which is always 2 π , to the radius, 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] .

Questions & Answers

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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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