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High School Math : The Unit Circle and Radians

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Example Questions

Example Question #31 : The Unit Circle And Radians

How many degrees are in $3\pi$ radians?

Possible Answers:

$460^\circ$

$510^\circ$

$340^\circ$

$540^\circ$

$180^\circ$

Correct answer:

$540^\circ$

Explanation:

The ratio between radians and degrees is $180^\circ=\pi$ radians.

That means we can set up a proportion:

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

$x\pi=180^\circ*3\pi$

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

$x=180^\circ*3$

$x=540^\circ$

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Example Question #52 : Trigonometry

How many degrees are in $\frac{2}{3}\pi$ radians?

Possible Answers:

$150^\circ$

$120^\circ$

$240^\circ$

$60^\circ$

$30^\circ$

Correct answer:

$120^\circ$

Explanation:

The conversion rate for degrees and radians is $\frac{180^\circ}{\pi \: rad}$ .

Set up a proportion to solve.

$\frac{180^\circ}{\pi \: rad}=\frac{x^\circ}{\frac{2}{3}\pi \: rad}$

Cross multiply.

$180^\circ*\frac{2}{3}\pi \: rad=\pi \: rad *x^\circ$

Notice that the $\pi$ cancels out.

$180^\circ*\frac{2}{3}=x^\circ$

$120^\circ=x$

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Example Question #23 : Understanding Radians And Conversions

How many degrees are in $2.42\pi$ radians?

Possible Answers:

$74.39^\circ$

$435.6^\circ$

$233.67^\circ$

$217.8^\circ$

$148.76^\circ$

Correct answer:

$217.8^\circ$

Explanation:

The ratio of degrees to radians is $\frac{180^\circ}{2\pi}$ . Set up a proportion and solve.

$\frac{x^\circ}{2.42\pi}=\frac{180^\circ}{2\pi}$

$x^\circ*2\pi=180^\circ*2.42\pi$

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

$x^\circ*2=180^\circ*2.42$

$x^\circ*2=435.6^\circ$

$x^\circ=\frac{435.6^\circ}{2}$

$x^\circ=217.8^\circ$

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Example Question #32 : The Unit Circle And Radians

How many radians are in $512^\circ$ ?

Possible Answers:

$1.42\pi$

$0.35\pi$

$2.84\pi$

$1.84\pi$

$0.70\pi$

Correct answer:

$2.84\pi$

Explanation:

The relationship between degrees and radians is $180^\circ=\pi\, rads$ . We can set this up as a proportion to solve:

$\frac{512^\circ}{x}=\frac{180^\circ}{\pi}$

$512^\circ*\pi=180^\circ*x$

$\frac{512^\circ}{180^\circ}\pi=x$

$2.84\pi=x$

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Example Question #31 : The Unit Circle And Radians

Convert $315^{\circ}$ to radians.

Possible Answers:

$\frac{7 }{4}$

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

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

$\frac{10\pi }{4}$

$\frac{11\pi }{4}$

Correct answer:

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

Explanation:

Multiply the given degrees measurement by $\frac{\pi }{180}$ .

Simplify and you get $\frac{7\pi }{4}$ .

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Example Question #61 : Trigonometry

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

Possible Answers:

$160^{\circ}$

$380^{\circ}$

$560^{\circ}$

$360^{\circ}$

$480^{\circ}$

Correct answer:

$480^{\circ}$

Explanation:

Multiply by $\frac{180}{\pi }$ and then simplify. The answer is $480^{\circ}$ .

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Example Question #23 : Understanding Radians And Conversions

How many radians are in $230^\circ$ ?

Possible Answers:

$2.46\: rad$

$6.44\: rad$

$2.56\: rad$

$0.81\: rad$

$8.03\: rad$

Correct answer:

$8.03\: rad$

Explanation:

The ratio of degrees to radians is $\frac{180^\circ}{2\pi \,radians}$ . Set up a proportion and solve.

$\frac{230^\circ}{x}=\frac{180^\circ}{2\pi \, rad}$

$230^\circ*2\pi \ rad=180^\circ*x$

$\frac{230^\circ*2\pi \ rad}{180^\circ}=x$

$2.56\pi \ rad=x$

$8.03\ rad=x$

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Example Question #28 : Understanding Radians And Conversions

How many degrees are in $\frac{5}{3}\pi$ radians?

Possible Answers:

$300^\circ$

$216^\circ$

$108^\circ$

$54^\circ$

$150^\circ$

Correct answer:

$300^\circ$

Explanation:

The conversion between radians and degrees is $\pi={180^\circ}$ . We can use this as a ratio to solve:

$\frac{\frac{5}{3}\pi}{x^\circ}=\frac{\pi}{180^\circ}$

Cross multipy.

$x^\circ*\pi=\frac{5}{3}\pi *180^\circ$

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

$x^\circ=\frac{5}{3} *180^\circ$

$x^\circ=300^\circ$

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Example Question #29 : Understanding Radians And Conversions

How many degrees are in $\frac{2}{3}\pi$ radians?

Possible Answers:

$120^\circ$

$180^\circ$

$270^\circ$

$60^\circ$

$240^\circ$

Correct answer:

$120^\circ$

Explanation:

The conversion between radians and degrees is $\pi={180^\circ}$ . We can then create a ratio to solve:

$\frac{\frac{2}{3}\pi}{x^\circ}=\frac{\pi}{180^\circ}$

Cross multipy:

$x^\circ*\pi=\frac{2}{3}\pi *180^\circ$

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

$x^\circ=\frac{2}{3} *180^\circ$

$x^\circ=120^\circ$

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Example Question #30 : Understanding Radians And Conversions

If Angle equals $72^{\circ}$ , what is the equivalent angle in radians (to the nearest hundredth)?

Possible Answers:

$4125.29\ radians$

$0.628\ radians$

$1.26\ radians$

$1256.6\ radians$

$4.125\ radians$

Correct answer:

$1.26\ radians$

Explanation:

To convert between radians and degrees, it is important to remember that:

$\pi\, rad = 180^{\circ}$

With this relationship in mind, we can convert from degrees to radians with the following formula:

$72^{\circ}\cdot \frac{\pi}{180^{\circ}} = 1.26\,rad$

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High School Math : The Unit Circle and Radians

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #31 : The Unit Circle And Radians

Example Question #52 : Trigonometry

Example Question #23 : Understanding Radians And Conversions

Example Question #32 : The Unit Circle And Radians

Example Question #31 : The Unit Circle And Radians

Example Question #61 : Trigonometry

Example Question #23 : Understanding Radians And Conversions

Example Question #28 : Understanding Radians And Conversions

Example Question #29 : Understanding Radians And Conversions

Example Question #30 : Understanding Radians And Conversions

All High School Math Resources

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