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Precalculus : Find the Degree Measure of an Angle

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Example Questions

Example Question #1 : Angles In The Coordinate Plane

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

Possible Answers:

360 $\displaystyle 360$

860 $\displaystyle 860$

240 $\displaystyle 240$

900 $\displaystyle 900$

480 $\displaystyle 480$

Correct answer:

480 $\displaystyle 480$

Explanation:

Write the conversion factor between radians and degrees.

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

Cancel the radians unit by using dimensional analysis.

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

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

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

Possible Answers:

10.8 $\displaystyle 10.8$

21.6 $\displaystyle 21.6$

$\displaystyle 48$

3.8 $\displaystyle 3.8$

16.4 $\displaystyle 16.4$

Correct answer:

10.8 $\displaystyle 10.8$

Explanation:

Write the conversion factor of radians and degrees.

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

Substitute the degree measure into $\pi$ $\displaystyle \pi$.

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

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Example Question #295 : Pre Calculus

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

Assume $\theta >0$ $\displaystyle \theta >0$

Possible Answers:

$\theta= 240^o$ $\displaystyle \theta= 240^o$

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

$\theta= 225^o$ $\displaystyle \theta= 225^o$

$\theta= 135^o$ $\displaystyle \theta= 135^o$

Correct answer:

$\theta= 225^o$ $\displaystyle \theta= 225^o$

Explanation:

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

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

$\gamma=45^o$ $\displaystyle \gamma=45^o$

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

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

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Example Question #296 : Pre Calculus

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

Possible Answers:

248 $\displaystyle 248$

237 $\displaystyle 237$

242 $\displaystyle 242$

215 $\displaystyle 215$

258 $\displaystyle 258$

Correct answer:

248 $\displaystyle 248$

Explanation:

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

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

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

The nearest degree is 248 $\displaystyle 248$.

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Example Question #297 : Pre Calculus

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

Possible Answers:

120 $\displaystyle 120$

$\displaystyle 30$

$\displaystyle 45$

$\displaystyle 90$

$\displaystyle 60$

Correct answer:

$\displaystyle 60$

Explanation:

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

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

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

The third angle is 60 degrees.

Let the second angle be $\displaystyle x$. The first angle three times the value of the second angle is $\displaystyle 3x$. Set up an equation that sums the three angles to $180 \textup{ degrees}$ $\displaystyle 180 \textup{ degrees}$.

$\displaystyle 3x+x+60 = 180$

Solve for $\displaystyle x$.

$\displaystyle 4x+60 = 180$

4x=120 $\displaystyle 4x=120$

x=30 $\displaystyle x=30$

Substitute x=30 $\displaystyle x=30$ for the first angle and second angle.

The second angle is: x=30 $\displaystyle x=30$

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

The three angles are: $\displaystyle 30,60,90$

The second highest angle is: $\displaystyle 60$

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Precalculus : Find the Degree Measure of an Angle

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Angles In The Coordinate Plane

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

Example Question #295 : Pre Calculus

Example Question #296 : Pre Calculus

Example Question #297 : Pre Calculus

All Precalculus Resources

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