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High School Math : Angles

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Example Questions

Example Question #1 : Understanding Coterminal Angles

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

Possible Answers:

$300^{\circ}$

$360^{\circ}$

$-15^{\circ}$

$85^{\circ}$

$305^{\circ}$

Correct answer:

$305^{\circ}$

Explanation:

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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Example Question #2 : Understanding Coterminal Angles

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

Possible Answers:

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

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

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

Each angle given in the other choices is coterminal with $\frac{5\pi}{9}$ .

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

Correct answer:

Each angle given in the other choices is coterminal with $\frac{5\pi}{9}$ .

Explanation:

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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Example Question #3 : Understanding Coterminal Angles

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

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

Possible Answers:

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

$2\pi$

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

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

$\pi$

Correct answer:

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

Explanation:

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

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

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

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Example Question #1 : Understanding Coterminal Angles

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

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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Example Question #21 : Trigonometry

Which one of these is positive in quadrant III?

Possible Answers:

Cosine

Sine

All trig functions

Tangent

No trig functions

Correct answer:

Tangent

Explanation:

The pattern for positive functions is All Student Take Calculus. In quandrant I, all trigonometric functions are positive. In quadrant II, sine is positive. In qudrant III, tangent is positive. In quadrant IV, cosine is positive.

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High School Math : Angles

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Understanding Coterminal Angles

Example Question #2 : Understanding Coterminal Angles

Example Question #3 : Understanding Coterminal Angles

Example Question #1 : Understanding Coterminal Angles

Example Question #21 : Trigonometry

All High School Math Resources

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