Find an angle
$\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ that is coterminal with an angle measuring
$\text{\hspace{0.17em}}\mathrm{870\xb0},$ where
$\text{\hspace{0.17em}}\mathrm{0\xb0}\le \alpha <\mathrm{360\xb0}.$
Given an angle with measure less than
$\text{\hspace{0.17em}}\mathrm{0\xb0},$ find a coterminal angle having a measure between
$\text{\hspace{0.17em}}\mathrm{0\xb0}\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}\mathrm{360\xb0}.$
Add
$\text{\hspace{0.17em}}\mathrm{360\xb0}\text{\hspace{0.17em}}$ to the given angle.
If the result is still less than
$\text{\hspace{0.17em}}\mathrm{0\xb0},$ add
$\text{\hspace{0.17em}}\mathrm{360\xb0}\text{\hspace{0.17em}}$ again until the result is between
$\text{\hspace{0.17em}}\mathrm{0\xb0}\text{\hspace{0.17em}}$ and
$\text{\hspace{0.17em}}\mathrm{360\xb0}.$
The resulting angle is coterminal with the original angle.
Finding an angle coterminal with an angle measuring less than
$\text{\hspace{0.17em}}\mathrm{0\xb0}$
Show the angle with measure
$\text{\hspace{0.17em}}\mathrm{-45\xb0}\text{\hspace{0.17em}}$ on a circle and find a positive coterminal angle
$\text{\hspace{0.17em}}\alpha \text{\hspace{0.17em}}$ such that
$\text{\hspace{0.17em}}\mathrm{0\xb0}\le \alpha <\mathrm{360\xb0}.$
Since
$\text{\hspace{0.17em}}\mathrm{45\xb0}\text{\hspace{0.17em}}$ is half of
$\text{\hspace{0.17em}}\mathrm{90\xb0},$ we can start at the positive horizontal axis and measure clockwise half of a
$\text{\hspace{0.17em}}\mathrm{90\xb0}\text{\hspace{0.17em}}$ angle.
Because we can find coterminal angles by adding or subtracting a full rotation of
$\text{\hspace{0.17em}}\mathrm{360\xb0},$ we can find a positive coterminal angle here by adding
$\text{\hspace{0.17em}}\mathrm{360\xb0}.$
Find an angle
$\text{\hspace{0.17em}}\beta \text{\hspace{0.17em}}$ that is coterminal with an angle measuring
$\text{\hspace{0.17em}}\mathrm{-300\xb0}\text{\hspace{0.17em}}$ such that
$\text{\hspace{0.17em}}\mathrm{0\xb0}\le \beta <\mathrm{360\xb0}.$
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
$\text{\hspace{0.17em}}2\pi .$
Subtract
$\text{\hspace{0.17em}}2\pi \text{\hspace{0.17em}}$ from the given angle.
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
$\text{\hspace{0.17em}}2\pi .$
The resulting angle is coterminal with the original angle.
Finding coterminal angles using radians
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:
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.
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
$\text{\hspace{0.17em}}0\le \theta <2\pi .$
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
$s=r\theta $
Given a circle of radius
$\text{\hspace{0.17em}}r,$ calculate the length
$\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ of the arc subtended by a given angle of measure
$\text{\hspace{0.17em}}\theta .$
If necessary, convert
$\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ to radians.
Multiply the radius
$\text{\hspace{0.17em}}r\text{\hspace{0.17em}}\text{\hspace{0.17em}}\theta :s=r\theta .$
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.
In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day?
Use your answer from part (a) to determine the radian measure for Mercury’s movement in one Earth day.
Let’s begin by finding the circumference of Mercury’s orbit.
what is the formula used for this question? "Jamal wants to save $54,000 for a down payment on a home. How much will he need to invest in an account with 8.2% APR, compounding daily, in order to reach his goal in 5 years?"