Angles  (Page 6/29)

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Find an angle $\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ that is coterminal with an angle measuring 870°, where $0°\le \alpha <360°.$

$\alpha =150°$

Given an angle with measure less than 0°, find a coterminal angle having a measure between 0° and 360°.

1. Add 360° to the given angle.
2. If the result is still less than 0°, add 360° again until the result is between 0° and 360°.
3. The resulting angle is coterminal with the original angle.

Finding an angle coterminal with an angle measuring less than 0°

Show the angle with measure −45° on a circle and find a positive coterminal angle $\alpha$ such that 0° ≤ α <360°.

Since 45° is half of 90°, we can start at the positive horizontal axis and measure clockwise half of a 90° angle.

Because we can find coterminal angles by adding or subtracting a full rotation of 360°, we can find a positive coterminal angle here by adding 360°:

$\text{\hspace{0.17em}}-45°+360°=315°\text{\hspace{0.17em}}$

We can then show the angle on a circle, as in [link] .

Find an angle $\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}$ that is coterminal with an angle measuring −300° such that $\text{\hspace{0.17em}}0°\le \beta <360°.\text{\hspace{0.17em}}$

$\beta =60°\text{\hspace{0.17em}}$

Finding coterminal angles measured in radians

We can find coterminal angles    measured in radians in much the same way as we have found them using degrees. In both cases, we find coterminal angles by adding or subtracting one or more full rotations.

Given an angle greater than $\text{\hspace{0.17em}}2\pi ,$ find a coterminal angle between 0 and $\text{\hspace{0.17em}}2\pi .$

1. Subtract $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ from the given angle.
2. If the result is still greater than $\text{\hspace{0.17em}}2\pi ,$ subtract $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ again until the result is between $\text{\hspace{0.17em}}0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}2\pi .\text{\hspace{0.17em}}$
3. The resulting angle is coterminal with the original angle.

Find an angle $\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}$ that is coterminal with $\text{\hspace{0.17em}}\frac{19\pi }{4},$ where $\text{\hspace{0.17em}}0\le \beta <2\pi .$

When working in degrees, we found coterminal angles by adding or subtracting 360 degrees, a full rotation. Likewise, in radians, we can find coterminal angles by adding or subtracting full rotations of $\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ radians:

$\begin{array}{l}\frac{19\pi }{4}-2\pi =\frac{19\pi }{4}-\frac{8\pi }{4}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{11\pi }{4}\hfill \end{array}$

The angle $\text{\hspace{0.17em}}\frac{11\pi }{4}\text{\hspace{0.17em}}$ is coterminal, but not less than $\text{\hspace{0.17em}}2\pi ,$ so we subtract another rotation:

$\begin{array}{l}\frac{11\pi }{4}-2\pi =\frac{11\pi }{4}-\frac{8\pi }{4}\hfill \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\frac{3\pi }{4}\hfill \end{array}$

The angle $\text{\hspace{0.17em}}\frac{3\pi }{4}\text{\hspace{0.17em}}$ is coterminal with $\text{\hspace{0.17em}}\frac{19\pi }{4},$ as shown in [link] .

Find an angle of measure $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ that is coterminal with an angle of measure $\text{\hspace{0.17em}}-\frac{17\pi }{6}\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}0\le \theta <2\pi .$

$\text{\hspace{0.17em}}\frac{7\pi }{6}\text{\hspace{0.17em}}$

Determining the length of an arc

Recall that the radian measure     $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ of an angle was defined as the ratio of the arc length     $\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ of a circular arc to the radius $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ of the circle, $\text{\hspace{0.17em}}\theta =\frac{s}{r}.\text{\hspace{0.17em}}$ From this relationship, we can find arc length along a circle, given an angle.

Arc length on a circle

In a circle of radius r , the length of an arc $\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ subtended by an angle with measure $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ in radians, shown in [link] , is

$\text{\hspace{0.17em}}s=r\theta \text{\hspace{0.17em}}$

Given a circle of radius $r,$ calculate the length $s$ of the arc subtended by a given angle of measure $\theta .$

1. If necessary, convert $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ to radians.
2. Multiply the radius $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ by the radian measure of $\text{\hspace{0.17em}}\theta :s=r\theta .\text{\hspace{0.17em}}$

Finding the length of an arc

Assume the orbit of Mercury around the sun is a perfect circle. Mercury is approximately 36 million miles from the sun.

1. In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day?
2. Use your answer from part (a) to determine the radian measure for Mercury’s movement in one Earth day.
1. Let’s begin by finding the circumference of Mercury’s orbit.

Since Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled:

2. Now, we convert to radians:

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