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Trigonometry : Angles

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Example Question #1 : Coterminal Angles

Which of the following angles is coterminal to $-130 ^{\circ}$ ?

Possible Answers:

$-230^{\circ}$

$130^{\circ}$

$230^{\circ}$

$50^{\circ}$

$-40^{\circ}$

Correct answer:

$230^{\circ}$

Explanation:

Coterminal angles share a common side when both are in standard position.

Thus, to find a coterminal angle for $-130 ^{\circ}$ , add $360^{\circ}$ to it to get a coterminal angle of $230^{\circ}$ .

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Example Question #1 : Coterminal Angles

Determine the next nearest positive and nearest negative coterminal angle to $\frac{\pi}{2}$

Possible Answers:

$\frac{\pi}{2} \hspace{0.1 in} and \hspace{0.1 in} -\frac{3\pi}{2}$

$\frac{5\pi}{2} \hspace{0.1 in} and \hspace{0.1 in} -\frac{\pi}{2}$

$\pi \hspace{0.1 in} and \hspace{0.1 in} -\frac{\pi}{2}$

$\frac{3\pi}{2} \hspace{0.1 in} and \hspace{0.1 in} -\frac{5\pi}{2}$

$\frac{5\pi}{2} \hspace{0.1 in} and \hspace{0.1 in} -\frac{3\pi}{2}$

Correct answer:

$\frac{5\pi}{2} \hspace{0.1 in} and \hspace{0.1 in} -\frac{3\pi}{2}$

Explanation:

In order to find a coterminal angle we must either subtract or add multiples of $2\pi$ . To find the nearest positive coterminal angle we add $2\pi$ :

$\frac{\pi}{2} + 2\pi = \frac{\pi}{2} + \frac{4\pi}{2} = \frac{5\pi}{2}$

To determine the nearest negative coterminal angle we subtract $2\pi$ :

$\frac{\pi}{2} - 2\pi = \frac{\pi}{2} - \frac{4\pi}{2} = -\frac{3\pi}{2}$

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Example Question #1 : Coterminal Angles

Determine the nearest negative coterminal angle to $\frac{7\pi}{9}$

Possible Answers:

$-\frac{11\pi}{9}$

$-\frac{2\pi}{9}$

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

$\frac{2\pi}{9}$

$\frac{11\pi}{9}$

Correct answer:

$-\frac{11\pi}{9}$

Explanation:

In order to determine a coterminal angle we add or subtract a multiple of $2\pi$ . In this case since the angle is positive and we want the nearest negative coterminal angle, we subtract $2\pi$ until we obtain the first negative angle as follows:

$\frac{7\pi}{9} - 2\pi = \frac{7\pi}{9} - \frac{18\pi}{9} = -\frac{11\pi}{9}$

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Example Question #1 : Coterminal Angles

Find the least positive coterminal angle to $-\frac{2\pi}{5}$

Possible Answers:

$\frac{3\pi]}{5}$

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

$\frac{8\pi}{5}$

$-\frac{3\pi}{5}$

$-\frac{8\pi}{5}$

Correct answer:

$\frac{8\pi}{5}$

Explanation:

To determine a coterminal angle we must add $2\pi$ to the original angle. In order to find the smallest positive coterminal angle, we add until we obtain the first positive angle. We first need to find the least common denominator as follows:

$-\frac{2\pi}{5} + 2\pi = -\frac{2\pi}{5} + \frac{10\pi}{5} = \frac{8\pi}{5}$

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Example Question #10 : Coterminal Angles

Find a positive and negative angle that are coterminal with a $60^{\circ}$ angle.

Possible Answers:

$300^{\circ}\ and\ -420^{\circ}$

$-300^{\circ}\ and\ 420^{\circ}$

$-300^{\circ}\ and\ 400^{\circ}$

$-330^{\circ}\ and\ 420^{\circ}$

$-360^{\circ}\ and\ 420^{\circ}$

Correct answer:

$-300^{\circ}\ and\ 420^{\circ}$

Explanation:

Coterminal angles are angles in standard position that have a common terminal side. In order to find a positive and a negative angle coterminal with $60^{\circ}$ , we can add and subtract $360^{\circ}$ . So we can write:

$60^{\circ}+360^{\circ}=420^{\circ}$

$60^{\circ}-360^{\circ}=-300^{\circ}$

So a $-300^{\circ}$ angle and a $420^{\circ}$ angle are coterminal with a $60^{\circ}$ angle.

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Example Question #41 : Angles

Find a positive and negative angle that are coterminal with a $\frac{2\pi}{5}$ angle.

Possible Answers:

$-\frac{8\pi}{5}\ and\ \frac{12\pi}{5}$

$-\frac{8\pi}{5}\ and\ \frac{\pi}{5}$

$-\frac{8\pi}{5}\ and\ \frac{11\pi}{5}$

$-\frac{6\pi}{5}\ and\ \frac{12\pi}{5}$

$-\frac{7\pi}{5}\ and\ \frac{12\pi}{5}$

Correct answer:

$-\frac{8\pi}{5}\ and\ \frac{12\pi}{5}$

Explanation:

Coterminal angles are angles in standard position that have a common terminal side. In order to find a positive and a negative angle coterminal with $\frac{2\pi}{5}$ , we can add and subtract $2\pi$ . So we can write:

$\frac{2\pi}{5}-2\pi=\frac{2\pi-10\pi}{5}=-\frac{8\pi}{5}$

$\frac{2\pi}{5}+2\pi=\frac{2\pi+10\pi}{5}=\frac{12\pi}{5}$

So a $-\frac{8\pi}{5}$ angle and a $\frac{12\pi}{5}$ angle are coterminal with a $\frac{2\pi}{5}$ angle.

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Example Question #12 : Coterminal Angles

Find a positive and negative angle that are coterminal with a $440^{\circ}$ angle.

Possible Answers:

$-280^{\circ}\ and \ 300^{\circ}$

$-280^{\circ}\ and \ 280^{\circ}$

$-260^{\circ}\ and \ 80^{\circ}$

$-280^{\circ}\ and \ 80^{\circ}$

$-380^{\circ}\ and \ 380^{\circ}$

Correct answer:

$-280^{\circ}\ and \ 80^{\circ}$

Explanation:

Coterminal angles are angles in standard position that have a common terminal side. In order to find a positive and a negative angle coterminal with $440^{\circ}$ , we need to subtract one full rotation ( $360^{\circ}$ ) and two full rotations ( $2\times 360^{\circ}$ ):

$440^{\circ}-360^{\circ}=80^{\circ}$

$440^{\circ}-2(360)^{\circ}=-280^{\circ}$

So a $-280^{\circ}$ angle and a $80^{\circ}$ angle are coterminal with a $440^{\circ}$ angle.

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Example Question #13 : Coterminal Angles

Which of the following is a coterminal angle to $\frac{\pi}{5}$ ?

Possible Answers:

$\pi$

$\frac{3\pi}{4}$

$\frac{11\pi}{3}$

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

$-\frac{13\pi}{2}$

Correct answer:

$-\frac{13\pi}{2}$

Explanation:

To find a possible coterminal angle, add or subtract multiples of 360 degrees, or $2\pi$ radians. In this case, since we are given the angle in radians, add or subtract $2\pi$ .

$\frac{\pi}{5}+2\pi = \frac{11\pi}{2}$

$\frac{11\pi}{2}+2\pi= \frac{15\pi}{2}$

$\frac{\pi}{5}-2\pi = -\frac{9\pi}{2}$

$-\frac{9\pi}{2}-2\pi=-\frac{13\pi}{2}$

These are all possible coterminal angles.

The correct answer is: $-\frac{13\pi}{2}$

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Example Question #14 : Coterminal Angles

Which angle is NOT coterminal with $\small \frac{\pi}{3}$ ?

Possible Answers:

$\small \frac{11\pi}{3}$

$\small \frac{-5\pi}{3}$

$\small \frac{7\pi}{3}$

$\small \frac{19\pi}{3}$

$\small \frac{-11\pi}{3}$

Correct answer:

$\small \frac{11\pi}{3}$

Explanation:

To obtain any angle that is coterminal with $\small \frac{\pi}{3}$ , either add or subtract $2\pi$ , or in this case its equivalent, $\small \frac{6\pi}{3}$ .

Adding yields: $\small \frac{\pi}{3}+\frac{6\pi}{3} ={\color{DarkOrange} \frac{7\pi}{3}} + \frac{6\pi}{3} = \frac{13\pi}{3} + \frac{6\pi}{3}= {\color{DarkOrange} \frac{19\pi}{3}}$

Subtracting yields: $\small \frac{\pi}{3} - \frac{6\pi}{3} ={\color{DarkOrange} \frac{-5\pi}{3} }- \frac{6\pi}{3} ={\color{DarkOrange} \frac{-11\pi}{3} }$

The only answer not generated either way is $\small \frac{11\pi}{3}$ .

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Example Question #15 : Coterminal Angles

Which angle is NOT coterminal with $\small \frac{-3\pi}{4}$ ?

Possible Answers:

$\small \frac{13\pi}{4}$

$\small \frac{5\pi}{4}$

$\small \frac{-11\pi}{4}$

$\small \frac{3\pi}{4}$

$\small \frac{-19\pi}{4}$

Correct answer:

$\small \frac{3\pi}{4}$

Explanation:

To obtain any angle that is coterminal with $\small \small \frac{-3\pi}{4}$ , either add or subtract $2\pi$ , or in this case its equivalent, $\small \small \frac{8\pi}{4}$ .

Adding yields: $\small \frac{-3\pi}{4}+\frac{8\pi}{4} ={\color{DarkOrange} \frac{5\pi}{4}} + \frac{8\pi}{4} = {\color{DarkOrange} \frac{13\pi}{4} }$

Subtracting yields: $\small \frac{-3\pi}{4} - \frac{8\pi}{4} ={\color{DarkOrange} \frac{-11\pi}{4} }- \frac{8\pi}{4} ={\color{DarkOrange} \frac{-19\pi}{4} }$

The only answer not generated either way is $\small \small \frac{3\pi}{4}$ .

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Trigonometry : Angles

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 : Coterminal Angles

Example Question #1 : Coterminal Angles

Example Question #1 : Coterminal Angles

Example Question #1 : Coterminal Angles

Example Question #10 : Coterminal Angles

Example Question #41 : Angles

Example Question #12 : Coterminal Angles

Example Question #13 : Coterminal Angles

Example Question #14 : Coterminal Angles

Example Question #15 : Coterminal Angles

All Trigonometry Resources

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