5.1 Angles  (Page 2/29)

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 $\frac{\mathrm{90°}}{\mathrm{360°}}=\frac{1}{4}.$ 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 $\frac{\mathrm{360°}}{\mathrm{360°}}=1.$ 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] .

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] .

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Show an angle of 240° on a circle in standard position.

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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 $C=2\pi r.$ If we divide both sides of this equation by $r,$ we create the ratio of the circumference to the radius, which is always $2\pi$ regardless of the length of the radius. So the circumference of any circle is $2\pi \approx 6.28$ 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] .