Trigonometry - Coterminal Angles

Find the least positive coterminal angle to $-1140^{\circ}$ .

$300^{\circ}$

$-780^{\circ}$

$60^{\circ}$

$420^{\circ}$

Explanation

To find a coterminal angle, you must add or subtract $360^{\circ}$ . The question is asking for the least positive coterminal angle, so you must add $360^{\circ}$ until you reach a positive angle.

$-1140^{\circ} + 360^{\circ} = -780^{\circ}$

The angle is still negative, so you must continue.

$-780^{\circ} + 360^{\circ} = -420^{\circ}$

The angle is still negative, so you must continue.

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

The angle is still negative, so you must continue.

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

What is the coterminal angle of 82o?

-278o

-8o

-98o

-45o

-90o

Explanation

The coterminal angle is the negative angle that travels from $\small 2\pi$ from the original angle.

$360^{\circ} - 82^{\circ} = 278^{\circ}$

Find a positive and a negative angle coterminal with a $\frac{\pi}{4}$ angle.

$-\frac{7\pi}{4}\ and \ \frac{9\pi}{4}$

$-\frac{7\pi}{4}\ and \ \frac{7\pi}{4}$

$-\frac{9\pi}{4}\ and \ \frac{9\pi}{4}$

$-\frac{3\pi}{4}\ and \ \frac{9\pi}{4}$

$-\frac{7\pi}{4}\ and \ \frac{11\pi}{4}$

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 $\frac{\pi}{4}$ , we can add and subtract $2\pi$ . So we can write:

$\frac{\pi}{4}-2\pi=\frac{\pi-8\pi}{4}=-\frac{7\pi}{4}$

$\frac{\pi}{4}+2\pi=\frac{\pi+8\pi}{4}=\frac{9\pi}{4}$

So a $-\frac{7\pi}{4}$ angle and a $\frac{9\pi}{4}$ angle are coterminal with a $\frac{\pi}{4}$ angle.

Find a positive and a negative angle coterminal with a $3\pi$ angle.

$\pi \ and\ -\pi$

$\pi \ and\ -2\pi$

$2\pi \ and\ -\pi$

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

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

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 $3\pi$ , we need to subtract one full rotation ( $2\pi$ ) and two full rotations ( $2\times 2\pi$ ):

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

$3\pi-2\times 2\pi=-\pi$

So a $\pi$ angle and a $(-\pi)$ angle are coterminal with a $3\pi$ angle.

Identify coterminal angles among the following pairs:

$(60^{\circ},410^{\circ})$

$(50^{\circ},-310^{\circ})$

$(80^{\circ},280^{\circ})$

$(45^{\circ},135^{\circ})$

$(50^{\circ},230^{\circ})$

$(50^{\circ},-310^{\circ})$

$(60^{\circ},410^{\circ})$

$(50^{\circ},230^{\circ})$

$(45^{\circ},135^{\circ})$

$(80^{\circ},280^{\circ})$

Explanation

Coterminal angles are angles in standard position that have a common terminal side.

So of the given pairs only $(50^{\circ},-310^{\circ})$ is the correct pair.

In order to examine the given pairs we can add $360^{\circ}$ to an angle or subtract $360^{\circ}$ from an angle:

$50^{\circ}-360^{\circ}=-310^{\circ}\Rightarrow -310^{\circ}\ and \50^{\circ}\ are \ coterminal$

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

$230^{\circ}$

$130^{\circ}$

$50^{\circ}$

$-40^{\circ}$

$-230^{\circ}$

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 $230^{\circ}$ .

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

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

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

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

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

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

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 $2\pi$ :

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

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

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

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

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

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

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

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:

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

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

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

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

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

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

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

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:

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

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

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

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

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

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

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

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: