Find the Degree Measure of an Angle - Pre-Calculus

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Question

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

Answer

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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Question

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

Answer

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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Question

Determine the angle $\theta$ in degres made in the plane by connecting a line segment from the origin to .

Assume $\theta >0$

Answer

Firstly, since the point is in the 3rd quadrant, it'll be between and . If we take 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 to be the horizontal, the total angle measure $\theta$ is going to be:

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

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Question

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

Answer

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 .

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Question

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?

Answer

A triangle has three angles that will add up to 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 $180 \textup{ degrees}$ .

Solve for .

Substitute for the first angle and second angle.

The second angle is:

The first angle is:

The three angles are:

The second highest angle is:

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