Understanding Coterminal Angles - High School Math

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Question

Find a coterminal angle for $-55^{\circ}$ .

Answer

Coterminal angles are angles that, when drawn in the standard position, share a terminal side. You can find these angles by adding or subtracting 360 to the given angle. Thus, the only angle measurement that works from the answers given is $305^{\circ}$ .

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Question

Which of the following angles is coterminal with $\frac{2\pi}{7}$ ?

Answer

For an angle to be coterminal with $\theta$ , that angle must be of the form $\theta + 2\pi N$ for some integer - or, equivalently, the difference of the angle measures multiplied by $\frac{1}{2\pi}$ must be an integer. We apply this test to all five choices.

$\frac{9\pi}{7}$ :

$\left ( \frac{1}{2\pi } \right ) \left ( \frac{2\pi}{7} - \frac{9\pi}{7} \right )$

$=\left ( \frac{1}{2\pi } \right )\left (- \frac{7\pi}{7} \right ) =\left ( \frac{1}{2\pi } \right ) \pi = \frac{1}{2 }$

$\frac{12\pi}{7}$ :

$\left ( \frac{1}{2\pi } \right ) \left ( \frac{2\pi}{7} - \frac{12\pi}{7} \right )$

$=\left ( \frac{1}{2\pi } \right )\left (- \frac{10\pi}{7} \right ) =- \frac{5}{7}$

$-\frac{2\pi}{7}$ :

$\left ( \frac{1}{2\pi } \right ) \left [\frac{2\pi}{7} -\left ( -\frac{2\pi}{7} \right ) \right ]$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{2\pi}{7} + \frac{2\pi}{7 }\right )$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{4\pi}{7} \right ) = \frac{2}{7}$

$-\frac{9\pi}{7}$ :

$\left ( \frac{1}{2\pi } \right ) \left [\frac{2\pi}{7} -\left ( -\frac{9\pi}{7} \right ) \right ]$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{2\pi}{7} + \frac{9\pi}{7 }\right )$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{11\pi}{7} \right ) = \frac{11}{14 }$

$-\frac{12\pi}{7}$ :

$\left ( \frac{1}{2\pi } \right ) \left [\frac{2\pi}{7} -\left ( -\frac{12\pi}{7} \right ) \right ]$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{2\pi}{7} + \frac{12\pi}{7 }\right )$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{14\pi}{7} \right ) = \left ( \frac{1}{2\pi } \right ) \cdot2\pi = 1$

$-\frac{12\pi}{7}$ is the correct choice, since only that choice passes our test.

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Question

Which of the following angles is coterminal with $\frac{5\pi}{9}$ ?

Answer

For an angle to be coterminal with $\theta$ , that angle must be of the form $\theta + 2\pi N$ for some integer - or, equivalently, the difference of the angle measures multiplied by $\frac{1}{2\pi}$ must be an integer. We apply this test to all four choices.

$\frac{41\pi}{9}$ :

$\left ( \frac{1}{2\pi } \right ) \left (\frac{41\pi}{9}- \frac{5\pi}{9}\right )$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{36\pi}{9} \right ) =\left ( \frac{1}{2\pi } \right )4 \pi =2$

$\frac{23\pi}{9}$ :

$\left ( \frac{1}{2\pi } \right ) \left (\frac{23\pi}{9}- \frac{5\pi}{9}\right )$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{18\pi}{9} \right ) =\left ( \frac{1}{2\pi } \right )2 \pi =1$

$-\frac{13\pi}{9}$ :

$\left ( \frac{1}{2\pi } \right ) \left (-\frac{13\pi}{9}- \frac{5\pi}{9}\right )$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{-18\pi}{9} \right ) =\left ( \frac{1}{2\pi } \right )\left (- 2 \pi \right )=-1$

$-\frac{31\pi}{9}$ :

$\left ( \frac{1}{2\pi } \right ) \left (-\frac{31\pi}{9}- \frac{5\pi}{9}\right )$

$=\left ( \frac{1}{2\pi } \right )\left ( \frac{-36\pi}{9} \right ) =\left ( \frac{1}{2\pi } \right )\left (- 4 \pi \right )=-2$

All four choices pass the test, so all four angles are coterminal with $\frac{5\pi}{9}$ .

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Question

Which of the following choices represents a pair of coterminal angles?

Answer

For two angles to be coterminal, they must differ by $2\pi N$ for some integer - or, equivalently, the difference of the angle measures multiplied by $\frac{1}{2\pi}$ must be an integer. We apply this test to all five choices.

$\frac{17\pi }{2} \textrm{ and } \frac{23\pi }{2}$ :

$\frac{1}{2\pi } \cdot \left ( \frac{23\pi }{2} - \frac{17\pi }{2} \right ) = \frac{1}{2\pi } \cdot \frac{6\pi }{2} = \frac{3 }{2}$

$\frac{17\pi }{3} \textrm{ and } \frac{23\pi }{3}$ :

$\frac{1}{2\pi } \cdot \left ( \frac{23\pi }{3} - \frac{17\pi }{3} \right ) = \frac{1}{2\pi } \cdot \frac{6\pi }{3} = 1$

$\frac{17\pi }{4} \textrm{ and } \frac{23\pi }{4}$ :

$\frac{1}{2\pi } \cdot \left ( \frac{23\pi }{4} - \frac{17\pi }{4} \right ) = \frac{1}{2\pi } \cdot \frac{6\pi }{4} = \frac{3 }{4}$

$\frac{17\pi }{5} \textrm{ and } \frac{23\pi }{5}$ :

$\frac{1}{2\pi } \cdot \left ( \frac{23\pi }{5} - \frac{17\pi }{5} \right ) = \frac{1}{2\pi } \cdot \frac{6\pi }{5} = \frac{3 }{5}$

$\frac{17\pi }{6} \textrm{ and } \frac{23\pi }{6}$ :

$\frac{1}{2\pi } \cdot \left ( \frac{23\pi }{6} - \frac{17\pi }{6} \right ) = \frac{1}{2\pi } \cdot \frac{6\pi }{6} = \frac{1 }{2}$

The only angles that pass the test - and are therefore coterminal - are $\frac{17\pi }{3} \textrm{ and } \frac{23\pi }{3}$ .

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Question

$\textup{Which of the following angle measures, given in radians, is coterminal with an}$

$\textup{an angle of 90 degrees?}$

Answer

$90\textup{ degrees }=\frac{\pi }{2}:: \textup{ Coterminal angles share the same position on the unit circle.}$

$\textup{Therefore, coterminal angles can be found by adding or subtracting multiples of }2\pi$ .

$\frac{\pi }{2}+2\pi=\frac{5\pi }{2}$

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