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Trigonometry : Coterminal Angles

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Example Questions

Example Question #41 : Angles

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

Possible Answers:

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

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

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

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

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

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 #11 : Coterminal Angles

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

Possible Answers:

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

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

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

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

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

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:

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

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

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

$\pi$

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

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 #11 : Coterminal Angles

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

Possible Answers:

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

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

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

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

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

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 #11 : Coterminal Angles

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

Possible Answers:

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

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

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

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

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

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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Example Question #12 : Coterminal Angles

Which two angles are both coterminal with $\small \frac{\pi}{2}$ ?

Possible Answers:

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

$\small \frac{-3\pi}{2}$ and $\small \frac{9\pi}{2}$

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

$\small \frac{3\pi}{2}$ and $\frac{-7\pi}{2}$

$\small \frac{5\pi}{2}$ and $\small \frac{-5\pi}{2}$

Correct answer:

$\small \frac{-3\pi}{2}$ and $\small \frac{9\pi}{2}$

Explanation:

For an angle to to coterminal with $\frac{\pi}{2}$ , it must be equivalent to that angle, pointing up. The two angles that work are $\small \frac{-3\pi}{2}$ and $\small \frac{9\pi}{2}$ .

We can visualize why $\small \frac{-3\pi}{2}$ works in a couple different ways. If we know that the angle pointing straight down is $\small \frac{3\pi}{2}$ , we can see that the negative version of that would point straight up. We could also count clockwise around the circle three $\small \frac{\pi}{2}$ -angles, which would place us at $\small \frac{\pi}{2}$ .

Similarly, counting around the unit circle nine $\small \frac{\pi}{2}$ -angles would place us at $\small \frac{\pi}{2}$ . We could also subtract $\small 2\pi$ , or equivalently $\small \frac{4\pi}{2}$ , to figure out where the angle is within the unit circle: $\small \small \frac{9\pi}{2}- \frac{4\pi}{2}= \frac{5\pi}{2} - \frac{4\pi}{2}= \frac{1\pi}{2}$

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Example Question #11 : Coterminal Angles

Find a positive coterminal angle of 390°.

Possible Answers:

120°

30°

60°

90°

10°

Correct answer:

30°

Explanation:

Coterminal angles have the same initial and terminal sides. The simplest case is 180°. If you imagine this on a cartesian plane, it is simply the x-axis. The 180° on the positive y-axis side is coterminal with the 180° on the negative y-axis side and vice versa. For an angle of 390° we find its positive coterminal angle by subtracting 360°. This gives us the coterminal angle of 30°. 30° and 390° both have the same initial and terminal sides.

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Example Question #12 : Coterminal Angles

Find a negative coterminal angle for 380°.

Possible Answers:

-340°

-380°

-310°

-300°

-330°

Correct answer:

-340°

Explanation:

Coterminal angles have the same initial and terminal sides. The simplest case is 180°. If you imagine this on a cartesian plane, it is simply the x-axis. The 180° on the positive y-axis side is coterminal with the 180° on the negative y-axis side and vice versa. To find the negative coterminal angle of 380°, we must subtract an angle that has the same terminal and initial sides, but is larger than 380° (this ensures we get a negative coterminal angle).

To find the correct "amount of angle" to subtract, multiply 360 by multiples of 2 until you get an angle value that would give a negative angle when subtracted.

$360^{\circ}(2)=720^{\circ}$

$380^{\circ}-720^{\circ}=-340^{\circ}$

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Example Question #11 : Coterminal Angles

Which of the following angles is coterminal with $\frac{28\pi}{9}+\frac{85\pi}{18}$ ?

Possible Answers:

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

$\frac{-\pi}{6}$

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

$\frac{-7\pi}{18}$

Correct answer:

$\frac{-\pi}{6}$

Explanation:

Adding the two fractions together yields

$\begin{align*} \frac{28\pi}{9}+\frac{85\pi}{18}&=\frac{(56+85)\pi}{18}\\ &=\frac{141\pi}{18}\\ &=\frac{47\pi}{6} \end{align*}$

We can find angles coterminal to this by adding or subtracting multiples of $2\pi$ .
In this case:

$\frac{47\pi}{6}-4(2\pi)=\frac{(47-48)\pi}{6}=\frac{-\pi}{6}$ .

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Example Question #13 : Coterminal Angles

Find positive angles between $0^{\circ}$ and $360^{\circ}$ which are coterminal to $-\frac{\pi}{2}rad$ , $1180^{\circ}$ , and $\frac{9\pi}{4}rad$ .

Possible Answers:

$\frac{3\pi}{2}rad$ , $180^{\circ}$ , $\frac{\pi}{8}rad$

$\frac{\pi}{2}rad$ , $180^{\circ}$ , $\frac{3\pi}{4}rad$

$\frac{3\pi}{2}rad$ , $100^{\circ}$ , $\frac{\pi}{4}rad$

$\frac{\pi}{6}rad$ , $270^{\circ}$ , $\frac{\pi}{4}rad$

$\frac{3\pi}{4}rad$ , $120^{\circ}$ , $\frac{\pi}{4}rad$

Correct answer:

$\frac{3\pi}{2}rad$ , $100^{\circ}$ , $\frac{\pi}{4}rad$

Explanation:

"Coterminal angles" are those angles in standard position that have a common terminal side. To find an angle coterminal to another given angle, simply add or subtract $360^{\circ}$ (or $2\pi rad$ ) to the given angle measure. The problem restricts the desired coterminal angles to those which lie between $0^{\circ}$ and $360^{\circ}$ . Hence, for each given angle measure we must find the angle between $0^{\circ}$ and $360^{\circ}$ which is equivalent to that angle by adding or subtracting multiples of $360^{\circ}$ or $2\pi rad$ .

$-\frac{\pi}{2}rad$ is negative, so $-\frac{\pi}{2}rad+2\pi rad = \frac{3\pi}{2}rad$ is coterminal to $-\frac{\pi}{2}rad$ and lies between 0rad and $2\pi rad$ .

$1180^{\circ}$ is greater than $360^{\circ}$ , so $1180^{\circ}-360^{\circ}-360^{\circ}-360^{\circ}=100^{\circ}$ is coterminal to $1180^{\circ}$ and lies between $0^{\circ}$ and $360^{\circ}$ .

$\frac{9\pi}{4}rad$ is greater than $2\pi rad$ , so $\frac{9\pi}{4}rad-2\pi rad-2\pi rad = \frac{\pi}{4}rad$ is coterminal to $\frac{9\pi}{4}rad$ and lies between 0rad and $2\pi rad$ .

Hence, the positive angles between $0^{\circ}$ and $360^{\circ}$ which are coterminal to $-\frac{\pi}{2}rad$ , $1180^{\circ}$ , and $\frac{9\pi}{4}rad$ are $\frac{3\pi}{2}rad$ , $100^{\circ}$ , $\frac{\pi}{4}rad$ , respectively.

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Trigonometry : Coterminal Angles

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #41 : Angles

Example Question #11 : Coterminal Angles

Example Question #13 : Coterminal Angles

Example Question #11 : Coterminal Angles

Example Question #11 : Coterminal Angles

Example Question #12 : Coterminal Angles

Example Question #11 : Coterminal Angles

Example Question #12 : Coterminal Angles

Example Question #11 : Coterminal Angles

Example Question #13 : Coterminal Angles

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