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Precalculus : Angles in the Coordinate Plane

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Example Question #1 : Find The Degree Measure Of An Angle

Convert $\frac{8\pi}{3}$ radians to degrees.

Possible Answers:

240

860

900

480

360

Correct answer:

480

Explanation:

Write the conversion factor between radians and degrees.

$\pi \textup{ radians} = 180 \textup{ degrees}$

Cancel the radians unit by using dimensional analysis.

$(\frac{8\pi}{3} \textup{ radians })(\frac{180 \textup{ degrees}}{\pi \textup{ radians}}) = 480 \textup{ degrees}$

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Example Question #2 : Find The Degree Measure Of An Angle

Convert $\frac{3}{50}\pi$ to degrees.

Possible Answers:

3.8

16.4

21.6

10.8

Correct answer:

10.8

Explanation:

Write the conversion factor of radians and degrees.

$\pi \textup{ radians}= 180 \textup{ degrees}$

Substitute the degree measure into $\pi$ .

$\frac{3}{50}\pi= \frac{3}{50}(180)= 10.8$

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Example Question #3 : Find The Degree Measure Of An Angle

Determine the angle $\theta$ in degres made in the plane by connecting a line segment from the origin to (-1,-1) .

Assume $\theta >0$

Possible Answers:

$\theta= 240^o$

$\theta= 135^o$

$\theta= 225^o$

$\theta= 245^o$ $\theta= 45^o$

Correct answer:

$\theta= 225^o$

Explanation:

Firstly, since the point (-1,-1) is in the 3rd quadrant, it'll be between 180^o and 270^o . If we take 180^o to be the horizontal, we can form a triangle with base and leg of values and . Solving for the angle in the 3rd quadrant given by $\gamma$ ,

$tan(\gamma)=\frac{-1}{-1}=1$

$\gamma=45^o$

Since this angle is made by assuming 180^o to be the horizontal, the total angle measure $\theta$ is going to be:

$\theta=180^o+45^o=225^o$

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Example Question #4 : Find The Degree Measure Of An Angle

Find the degree measure of $\frac{11}{8} \pi$ radians. Round to the nearest integer.

Possible Answers:

215

242

258

237

248

Correct answer:

248

Explanation:

In order to solve for the degree measure from radians, replace the $\pi$ radians with 180 degrees.

$\pi \textup{ radians} = 180 \textup{ degrees}$

$\frac{11}{8} (180) = \frac{11\times 45}{2} = \frac{495}{2}= 247.5$

The nearest degree is 248 .

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Example Question #5 : Find The Degree Measure Of An Angle

Given a triangle, the first angle is three times the value of the second angle. The third angle is $\frac{\pi}{3} \textup{ radians}$ . What is the value of the second largest angle in degrees?

Possible Answers:

120

Correct answer:

Explanation:

A triangle has three angles that will add up to 180 degrees.

Convert the radians angle to degrees by substituting $180 \textup{ degrees}$ for every $\pi \textup{ radian}$ .

$\frac{180}{3} = 60 \textup{ degrees}$

The third angle is 60 degrees.

Let the second angle be . The first angle three times the value of the second angle is . Set up an equation that sums the three angles to $180 \textup{ degrees}$ .

3x+x+60 = 180

Solve for .

4x+60 = 180

4x=120

x=30

Substitute x=30 for the first angle and second angle.

The second angle is: x=30

The first angle is: 3x= 3(30) = 90

The three angles are: 30,60,90

The second highest angle is:

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Example Question #1 : Find The Measure Of A Coterminal Angle

Find the coterminal angle of 15 degrees.

Possible Answers:

$-345^{\circ}$

$375^{\circ}$

$735^{\circ}$

$\textup{All of the answers represent coterminal angles.}$

$-705^{\circ}$

Correct answer:

$\textup{All of the answers represent coterminal angles.}$

Explanation:

The coterminal angles can be positive or negative. To find the coterminal angles, simply add or subtract 360 degrees as many times as needed from the reference angle.

15+360=375

15+360+360= 735

15-360=-345

15-360-360=-705

All of these angles are coterminal angles.

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Example Question #2 : Find The Measure Of A Coterminal Angle

Of the given answers, what of the following is a coterminal angle of $\pi$ radians?

Possible Answers:

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

$3\pi$

$2\pi$

$-2\pi$

$\frac{\pi}{2}$

Correct answer:

$3\pi$

Explanation:

To find the coterminal angle of an angle, simply add or subtract $2\pi$ radians, or 360 degrees as many times as needed.

$\pi+2\pi=3\pi$

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

$3\pi+2\pi=5\pi$

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

These are all coterminal angles to $\pi$ radians.

Out of the given answers, $3\pi$ is the only possible answer.

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Example Question #3 : Find The Measure Of A Coterminal Angle

Of the following choices, find a coterminal angle of $\pi$ .

Possible Answers:

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

$6\pi$

$-2\pi$

$-5\pi$

Correct answer:

$-5\pi$

Explanation:

In order to find a coterminal angle, simply add or subtract $2\pi$ radians to the given angle as many times as possible.

$\theta=\pi+2\pi k$

The possible angles after adding increments of $2\pi$ radians are:

$\pi, 3\pi,5\pi,7\pi...$

$\theta=\pi-2\pi k$

The possible angles after subtracting decrements of $2\pi$ radians are:

$\pi,-\pi,-3\pi,-5\pi...$

Out of the given possibilities, only $-5\pi$ is a valid answer.

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Example Question #4 : Find The Measure Of A Coterminal Angle

Find the coterminal angle of 15 degrees in standard position from the following answers.

Possible Answers:

$365^{\circ}$

$360^{\circ}$

$30^{\circ}$

$-705^{\circ}$

$-1060^{\circ}$

Correct answer:

$-705^{\circ}$

Explanation:

To determine the coterminal angle, simply add or subtract increments or decrements of 360 degrees to the given angle.

For n=1,2,3,... :

15+360n= 375,735,1095...

15-360n= -345,-705,-1065...

These angles can all be coterminal to 15 degrees. The only answer is .

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Example Question #5 : Find The Measure Of A Coterminal Angle

Find the coterminal angle of $10^\circ$ , if possible.

Possible Answers:

$350^\circ$

$-350^\circ$

$190^\circ$

$-190^\circ$

$\textup{The coterminal angle does not exist.}$

Correct answer:

$-350^\circ$

Explanation:

In order to find a coterminal angle, or angles of the given angle, simply add or subtract 360 degrees of the terminal angle as many times as possible.

10+360=370

10-360=-350

The only correct answer is .

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Precalculus : Angles in the Coordinate Plane

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Find The Degree Measure Of An Angle

Example Question #2 : Find The Degree Measure Of An Angle

Example Question #3 : Find The Degree Measure Of An Angle

Example Question #4 : Find The Degree Measure Of An Angle

Example Question #5 : Find The Degree Measure Of An Angle

Example Question #1 : Find The Measure Of A Coterminal Angle

Example Question #2 : Find The Measure Of A Coterminal Angle

Example Question #3 : Find The Measure Of A Coterminal Angle

Example Question #4 : Find The Measure Of A Coterminal Angle

Example Question #5 : Find The Measure Of A Coterminal Angle

All Precalculus Resources

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