7.1 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 $\text{\hspace{0.17em}}360°.\text{\hspace{0.17em}}$ For example, to draw a $\text{\hspace{0.17em}}90°\text{\hspace{0.17em}}$ angle, we calculate that $\text{\hspace{0.17em}}\frac{90°}{360°}=\frac{1}{4}.\text{\hspace{0.17em}}$ So, the terminal side will be one-fourth of the way around the circle, moving counterclockwise from the positive x -axis. To draw a $\text{\hspace{0.17em}}360°$ angle, we calculate that $\text{\hspace{0.17em}}\frac{360°}{360°}=1.\text{\hspace{0.17em}}$ 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 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 $\text{0°,}\text{\hspace{0.17em}}\text{90°,}\text{\hspace{0.17em}}\text{180°,}\text{\hspace{0.17em}}\text{270°,}$ or $\text{\hspace{0.17em}}\text{360°}.\text{\hspace{0.17em}}$ See [link] .

An angle is a quadrantal angle    if its terminal side lies on an axis, including $\text{0°,}\text{\hspace{0.17em}}\text{90°,}\text{\hspace{0.17em}}\text{180°,}\text{\hspace{0.17em}}\text{270°,}$ or $\text{\hspace{0.17em}}\text{360°}.$

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

1. Express the angle measure as a fraction of $\text{\hspace{0.17em}}\text{360°}.$
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 $\text{\hspace{0.17em}}30°\text{\hspace{0.17em}}$ in standard position.
2. Sketch an angle of $\text{\hspace{0.17em}}-135°\text{\hspace{0.17em}}$ in standard position.
1. Divide the angle measure by $\text{\hspace{0.17em}}360°.$

$\frac{30°}{360°}=\frac{1}{12}$

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

$\frac{1}{12}=\frac{1}{3}\left(\frac{1}{4}\right)$

One-twelfth equals one-third of a quarter, so by dividing a quarter rotation into thirds, we can sketch a line at $\text{\hspace{0.17em}}30°,$ as in [link] .

2. Divide the angle measure by $\text{\hspace{0.17em}}360°.$

$\frac{-135°}{360°}=-\frac{3}{8}$

In this case, we can recognize that

$-\frac{3}{8}=-\frac{3}{2}\left(\frac{1}{4}\right)$

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] .

Show an angle of $\text{\hspace{0.17em}}240°\text{\hspace{0.17em}}$ on a circle in standard position.

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 $\text{\hspace{0.17em}}C=2\pi r.\text{\hspace{0.17em}}$ If we divide both sides of this equation by $\text{\hspace{0.17em}}r,$ we create the ratio of the circumference, which is always $\text{\hspace{0.17em}}2\pi ,$ to the radius, regardless of the length of the radius. So the circumference of any circle is $\text{\hspace{0.17em}}2\pi \approx 6.28\text{\hspace{0.17em}}$ 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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