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Precalculus : Find the Measure of a Coterminal Angle

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Example Questions

Example Question #1 : Angles In The Coordinate Plane

Find the coterminal angle of 15 degrees.

Possible Answers:

$-345^{\circ}$ $\displaystyle -345^{\circ}$

$375^{\circ}$ $\displaystyle 375^{\circ}$

$-705^{\circ}$ $\displaystyle -705^{\circ}$

$735^{\circ}$ $\displaystyle 735^{\circ}$

$\textup{All of the answers represent coterminal angles.}$

Correct answer:

$\textup{All of the answers represent coterminal angles.}$

Explanation:

The coterminal angles can be positive or negative. To find the coterminal angles, simply add or subtract 360 degrees as many times as needed from the reference angle.

$\displaystyle 15+360=375$

$\displaystyle 15+360+360= 735$

$\displaystyle 15-360=-345$

$\displaystyle 15-360-360=-705$

All of these angles are coterminal angles.

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Example Question #1 : Find The Measure Of A Coterminal Angle

Of the given answers, what of the following is a coterminal angle of $\pi$ $\displaystyle \pi$ radians?

Possible Answers:

$\frac{3\pi}{2}$ $\displaystyle \frac{3\pi}{2}$

$\frac{\pi}{2}$ $\displaystyle \frac{\pi}{2}$

$3\pi$ $\displaystyle 3\pi$

$2\pi$ $\displaystyle 2\pi$

$-2\pi$ $\displaystyle -2\pi$

Correct answer:

$3\pi$ $\displaystyle 3\pi$

Explanation:

To find the coterminal angle of an angle, simply add or subtract $2\pi$ $\displaystyle 2\pi$ radians, or 360 degrees as many times as needed.

$\pi+2\pi=3\pi$ $\displaystyle \pi+2\pi=3\pi$

$\pi-2\pi=-\pi$ $\displaystyle \pi-2\pi=-\pi$

$3\pi+2\pi=5\pi$ $\displaystyle 3\pi+2\pi=5\pi$

$-\pi-2\pi=-3\pi$ $\displaystyle -\pi-2\pi=-3\pi$

These are all coterminal angles to $\pi$ $\displaystyle \pi$ radians.

Out of the given answers, $3\pi$ $\displaystyle 3\pi$ is the only possible answer.

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Example Question #2 : Find The Measure Of A Coterminal Angle

Of the following choices, find a coterminal angle of $\pi$ $\displaystyle \pi$ .

Possible Answers:

$\displaystyle 0$

$-2\pi$ $\displaystyle -2\pi$

$6\pi$ $\displaystyle 6\pi$

$-5\pi$ $\displaystyle -5\pi$

$\frac{3}{2}\pi$ $\displaystyle \frac{3}{2}\pi$

Correct answer:

$-5\pi$ $\displaystyle -5\pi$

Explanation:

In order to find a coterminal angle, simply add or subtract $2\pi$ $\displaystyle 2\pi$ radians to the given angle as many times as possible.

$\theta=\pi+2\pi k$ $\displaystyle \theta=\pi+2\pi k$

The possible angles after adding increments of $2\pi$ $\displaystyle 2\pi$ radians are:

$\pi, 3\pi,5\pi,7\pi...$ $\displaystyle \pi, 3\pi,5\pi,7\pi...$

$\theta=\pi-2\pi k$ $\displaystyle \theta=\pi-2\pi k$

The possible angles after subtracting decrements of $2\pi$ $\displaystyle 2\pi$ radians are:

$\pi,-\pi,-3\pi,-5\pi...$ $\displaystyle \pi,-\pi,-3\pi,-5\pi...$

Out of the given possibilities, only $-5\pi$ $\displaystyle -5\pi$ is a valid answer.

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Example Question #2 : Find The Measure Of A Coterminal Angle

Find the coterminal angle of 15 degrees in standard position from the following answers.

Possible Answers:

$365^{\circ}$ $\displaystyle 365^{\circ}$

$30^{\circ}$ $\displaystyle 30^{\circ}$

$360^{\circ}$ $\displaystyle 360^{\circ}$

$-1060^{\circ}$ $\displaystyle -1060^{\circ}$

$-705^{\circ}$ $\displaystyle -705^{\circ}$

Correct answer:

$-705^{\circ}$ $\displaystyle -705^{\circ}$

Explanation:

To determine the coterminal angle, simply add or subtract increments or decrements of 360 degrees to the given angle.

For $\displaystyle n=1,2,3,...$ :

$\displaystyle 15+360n= 375,735,1095...$

$\displaystyle 15-360n= -345,-705,-1065...$

These angles can all be coterminal to 15 degrees. The only answer is $\displaystyle -705$ .

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Example Question #1 : Find The Measure Of A Coterminal Angle

Find the coterminal angle of $10^\circ$ $\displaystyle 10^\circ$ , if possible.

Possible Answers:

$-190^\circ$ $\displaystyle -190^\circ$

$\textup{The coterminal angle does not exist.}$

$350^\circ$ $\displaystyle 350^\circ$

$190^\circ$ $\displaystyle 190^\circ$

$-350^\circ$ $\displaystyle -350^\circ$

Correct answer:

$-350^\circ$ $\displaystyle -350^\circ$

Explanation:

In order to find a coterminal angle, or angles of the given angle, simply add or subtract 360 degrees of the terminal angle as many times as possible.

$\displaystyle 10+360=370$

$\displaystyle 10-360=-350$

The only correct answer is $\displaystyle -350$ .

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Example Question #1 : Find The Measure Of A Coterminal Angle

Which of the following angles is coterminal to $65^{\circ}$ $\displaystyle 65^{\circ}$ ?

Possible Answers:

$-295^{\circ}$ $\displaystyle -295^{\circ}$

$-65^{\circ}$ $\displaystyle -65^{\circ}$

$-425^{\circ}$ $\displaystyle -425^{\circ}$

$295^{\circ}$ $\displaystyle 295^{\circ}$

Correct answer:

$-295^{\circ}$ $\displaystyle -295^{\circ}$

Explanation:

Which of the following angles is coterminal to $65^{\circ}$ $\displaystyle 65^{\circ}$ ?

Coterminal angles are angles which start and end at the same point. In other words, they share both their starting and ending point. Note, this doesn't require them to be the same angle.

For instance, $120^{\circ}$ $\displaystyle 120^{\circ}$ is coterminal with $-240^{\circ}$ $\displaystyle -240^{\circ}$ , because they both start on the positive x-axis, and end at the same place in quadrant 2.

So, we want to find an angle that ends at the same place in quadrant 1 as $65^{\circ}$ $\displaystyle 65^{\circ}$ . Of the answer choices, only 1 ends in quadrant 1, so that one must be our answer: $-295^{\circ}$ $\displaystyle -295^{\circ}$

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Precalculus : Find the Measure of a Coterminal Angle

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Angles In The Coordinate Plane

Example Question #1 : Find The Measure Of A Coterminal Angle

Example Question #2 : Find The Measure Of A Coterminal Angle

Example Question #2 : Find The Measure Of A Coterminal Angle

Example Question #1 : Find The Measure Of A Coterminal Angle

Example Question #1 : Find The Measure Of A Coterminal Angle

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