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Trigonometry : Angles

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Example Question #31 : Complementary And Supplementary Angles

Find the supplementary angle of $\frac{\pi}{4}$ :

Possible Answers:

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

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

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

$\frac{\pi}{2}$

$\frac{\pi}{4}$

Correct answer:

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

Explanation:

Since the given angle is in radians to find the supplementary angle we need to subtract the giving angle from $\pi$ .

Hence,

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

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Example Question #32 : Angles

Find the supplementary angle of $\frac{5\pi}{8}$

Possible Answers:

The supplementary angle of $\frac{5\pi}{8}$ does not exist, because $\frac{5\pi}{8}$ is larger than $\frac{\pi}{2}$ .

$\frac{11\pi}{8}$

$\frac{\pi}{8}$

$\frac{3\pi}{8}$

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

Correct answer:

$\frac{3\pi}{8}$

Explanation:

Since the given angle is in radians to find the supplementary angle we need to subtract the giving angle from $\pi$ .

Hence,

$\pi-\frac{5\pi}{8}=\frac{3\pi}{8}$

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Example Question #32 : Complementary And Supplementary Angles

What is the supplementary angle to $110\degree$ ?

Possible Answers:

$75\degree$

$60\degree$

$70\degree$

$40\degree$

Correct answer:

$70\degree$

Explanation:

Supplementary angles, by definition, add up to $180\degree$ .

To find the other angle, you set up the equation

$110\degree+x=180\degree$ .

Solving the equation gets

$x=70\degree$ .

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Example Question #34 : Angles

List the complement and the supplement for $\small \frac{\pi}{4}$ .

Possible Answers:

$\small \frac{3\pi}{4}$ and $\frac{7\pi}{4}$

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

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

$\small \small \frac{5\pi}{4}$ and $\small \frac{-3\pi}{4}$

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

Correct answer:

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

Explanation:

To find the complement of $\frac{\pi}{4}$ , subtract it from $\frac{\pi}{2}$ , or equivalently, $\frac{2\pi}{4}$ :

$\frac{2\pi}{4} - \frac{\pi}{4} = \frac{1\pi}{4}$

To find the supplement of $\frac{\pi}{4}$ , subtract it from $\pi$ , or equivalently $\frac{4\pi}{4}$ :

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

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Example Question #35 : Angles

Which angle is $\frac{2}{3}$ its supplement?

Possible Answers:

$\small \frac{3\pi}{10}$

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

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

$\frac{5\pi}{6}$

$\small \frac{6\pi}{5}$

Correct answer:

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

Explanation:

There are many ways to solve this problem. Regardless, we need to know that supplementary angles add together to $\pi$ . To figure out what plus $\frac{2}{3}$ of itself would equal $\pi$ , we can set up this equation:

$\small x + \frac{2}{3}x = \pi$ to combine like terms, create common denominators

$\small \frac{3}{3}x + \frac{2}{3}x = \pi$

$\small \frac{5}{3}x = \pi$ multiply both sides by the reciprocal, $\small \frac{3}{5}$

$\small x = \pi * \frac{3}{5 }$

$\small x = \frac{3\pi}{5}$

Our actual answer is going to be $\small \frac{2}{3}$ of this, since the angle that we are looking for is $\frac{2}{3 }$ of its supplement. In other words, we just found x, and the angle that we are looking for is $\small \frac{2}{3}x$ :

$\small \frac{2}{{\color{DarkOrange} 3}}*\frac{{\color{DarkOrange} 3}\pi}{5} = \frac{2}{1} * \frac{\pi}{5 }= \frac{2\pi}{5}$

Our answer is $\small \frac{2\pi}{5}$ .

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Example Question #1 : Coterminal Angles

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

Possible Answers:

$300^{\circ}$

$420^{\circ}$

$-780^{\circ}$

$60^{\circ}$

Correct answer:

$300^{\circ}$

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.

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

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Example Question #1 : Coterminal Angles

What is the coterminal angle of 82^o?

Possible Answers:

-278^o

-90^o

-45^o

-8^o

-98^o

Correct answer:

-278^o

Explanation:

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

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

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Example Question #3 : Coterminal Angles

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

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.

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Example Question #1 : Coterminal Angles

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

Possible Answers:

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

$\pi \ and\ -\pi$

$\pi \ and\ -2\pi$

$2\pi \ and\ -\pi$

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

Correct answer:

$\pi \ and\ -\pi$

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.

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Example Question #5 : Coterminal Angles

Identify coterminal angles among the following pairs:

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

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

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

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

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

Possible Answers:

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

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

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

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

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

Correct answer:

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

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:

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

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Trigonometry : Angles

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #31 : Complementary And Supplementary Angles

Example Question #32 : Angles

Example Question #32 : Complementary And Supplementary Angles

Example Question #34 : Angles

Example Question #35 : Angles

Example Question #1 : Coterminal Angles

Example Question #1 : Coterminal Angles

Example Question #3 : Coterminal Angles

Example Question #1 : Coterminal Angles

Example Question #5 : Coterminal Angles

All Trigonometry Resources

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