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High School Math : Understanding Radians and Conversions

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Example Question #1 : Understanding Radians And Conversions

What is the value of the angle $\frac{3\pi}{2}$ radians when converted to degrees?

Possible Answers:

270

135

Correct answer:

270

Explanation:

The answer can be found using the conversion of 1 radian equals $\frac{180}{\pi}$ degrees. Multiplying $\frac{3\pi}{2}$ by this conversion factor gives 270 degrees.

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Example Question #1 : Understanding Radians And Conversions

A point has Cartesian coordinates (1,3) . Rewrite this as an ordered pair in the polar coordinate plane, rounding the coordinates to the nearest hundredth.

Possible Answers:

(2.83, 0.32)

(2.83, 1.25)

(3.16,0.32)

(3.16,0.80)

(3.16,1.25)

Correct answer:

(3.16,1.25)

Explanation:

Set x=1, y=3 . Calculate the polar coordinates $(r,\theta )$ as follows:

$r = \sqrt{x^{2}+y^{2}}= \sqrt{1^{2}+3^{2}}=\sqrt{10} \approx 3.16$

$\theta = \arctan \frac{y}{x}= \arctan \frac{3}{1} = \arctan 3 \approx 1.25$

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Example Question #3 : Understanding Radians And Conversions

How many degrees are in $\frac{\pi}{6}$ radians?

Possible Answers:

$60^\circ$

$30^\circ$

$45^\circ$

$120^\circ$

$360^\circ$

Correct answer:

$30^\circ$

Explanation:

$\frac{x^\circ}{\frac{\pi}{6}}=\frac{180^\circ}{\pi}$

Cross multiply:

$x^\circ*\pi=180^\circ*\frac{\pi}{6}$

Notice that the $\pi$ 's cancel out:

$x^\circ=180^\circ*\frac{1}{6}$

$x^\circ=30^\circ$

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Example Question #1 : Understanding Radians And Conversions

How many radians are in $180^\circ$ ?

Possible Answers:

$3\pi$

$4\pi$

$\frac{\pi}{2}$

$\pi$

$2\pi$

Correct answer:

$\pi$

Explanation:

The relationship between degrees and radians is $180^\circ=\pi$ radians. Therefore, $180^\circ$ would be $\pi$ radians.

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Example Question #2 : Understanding Radians And Conversions

Express in radians: $125^{\circ }$

Possible Answers:

$\frac{15\pi }{36}$

$\frac{11\pi }{18}$

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

$\frac{25\pi }{36}$

$\frac{5\pi }{9}$

Correct answer:

$\frac{25\pi }{36}$

Explanation:

Since $180 ^{\circ } = \pi \textrm{ rad}$ , we can convert as follows:

$125 \cdot \frac{\pi }{180} = \frac{25\pi }{36}$

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Example Question #6 : Understanding Radians And Conversions

How many radians are in $85^\circ$ ?

Possible Answers:

$\frac{17}{36}\pi$

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

$\frac{36}{17}\pi$

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

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

Correct answer:

$\frac{17}{36}\pi$

Explanation:

Since there are $\pi$ radians for every $180^\circ$ , we can set up a proportion to solve.

$\frac{85^\circ}{x}=\frac{180^\circ}{\pi}$

Cross multiply.

$85^\circ*\pi=180^\circ *x$

Divide both sides by $180^\circ$ .

$\frac{85^\circ}{180^\circ}*\pi=x$

$\frac{17}{36}\pi=x$

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Example Question #7 : Understanding Radians And Conversions

How many radians are in $270^\circ$ ?

Possible Answers:

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

$2\pi$

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

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

$\pi$

Correct answer:

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

Explanation:

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{270^\circ}{x}=\frac{180^\circ}{\pi}$

Cross multiply:

$270^\circ*\pi=180^\circ*x$

Isolate :

$\frac{270^\circ}{180^\circ}\pi=x$

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

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Example Question #8 : Understanding Radians And Conversions

How many degrees are in $\frac{\pi}{3}$ radians?

Possible Answers:

$30^\circ$

$60^\circ$

$45^\circ$

$540^\circ$

$15^\circ$

Correct answer:

$60^\circ$

Explanation:

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{x^\circ}{\frac{\pi}{3}}=\frac{180^\circ}{\pi}$

Cross multiply:

$x^\circ*\pi=180^\circ*\frac{\pi}{3}$

Divide both sides by $\pi$ :

$x^\circ=\frac{180^\circ}{3}$

$x^\circ=60^\circ$

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Example Question #9 : Understanding Radians And Conversions

How many degrees are in $\frac{8\pi}{3}$ radians?

Possible Answers:

$420^\circ$

$480^\circ$

$120^\circ$

$30^\circ$

$460^\circ$

Correct answer:

$480^\circ$

Explanation:

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{x^\circ}{\frac{8\pi}{3}}=\frac{180^\circ}{\pi}$

Cross multiply:

$x^\circ*\pi=180^\circ*\frac{8\pi}{3}$

Notice that the $\pi$ 's cancel out:

$x^\circ=180^\circ*\frac{8}{3}$

$x^\circ=480^\circ$

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Example Question #10 : Understanding Radians And Conversions

How many degrees are in $\frac{3\pi}{4}$ radians?

Possible Answers:

$405^\circ$

$180^\circ$

$135^\circ$

$45^\circ$

$225^\circ$

Correct answer:

$135^\circ$

Explanation:

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{x^\circ}{\frac{3\pi}{4}}=\frac{180^\circ}{\pi}$

Cross multiply:

$x^\circ*\pi=180^\circ*\frac{3\pi}{4}$

Notice that the $\pi$ 's cancel out:

$x^\circ=180^\circ*\frac{3}{4}$

$x^\circ=135^\circ$

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High School Math : Understanding Radians and Conversions

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Understanding Radians And Conversions

Example Question #1 : Understanding Radians And Conversions

Example Question #3 : Understanding Radians And Conversions

Example Question #1 : Understanding Radians And Conversions

Example Question #2 : Understanding Radians And Conversions

Example Question #6 : Understanding Radians And Conversions

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