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Trigonometry : Unit Circle and Radians

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Example Questions

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Example Question #1 : Circles

Give in degrees:

$\frac{\frac{\pi}{2}-x}{\frac{\pi}{2}+x}=\frac{1}{2}$

Possible Answers:

$x=40^{\circ}$

$x=90^{\circ}$

$x=60^{\circ}$

$x=30^{\circ}$

$x=36^{\circ}$

Correct answer:

$x=30^{\circ}$

Explanation:

First we can find in radians:

$\frac{\frac{\pi}{2}-x}{\frac{\pi}{2}+x}=\frac{1}{2}\Rightarrow 2(\frac{\pi}{2}-x)=1(\frac{\pi}{2}+x)$

$\Rightarrow \pi-2x=\frac{\pi}{2}+x\Rightarrow \pi-\frac{\pi}{2}=3x$

$\Rightarrow 3x=\frac{\pi}{2}\Rightarrow x=\frac{\pi}{6}$

To change radians to degrees we need to multiply radians by $\frac{180^{\circ}}{\pi}$ . So we can write:

$x=\frac{\pi}{6}\Rightarrow x=\frac{\pi}{6}\times \frac{180^{\circ}}{\pi}=30^{\circ}$

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Example Question #1 : Circles

Convert the angle $\frac{7\pi }{3}$ into degrees.

Possible Answers:

$315^{\circ}$

$840^{\circ}$

$210^{\circ}$

$500^{\circ}$

$420^{\circ}$

Correct answer:

$420^{\circ}$

Explanation:

To convert radians to degrees, use the conversion $\frac{180^{\circ}}{\pi }$ .

In this case:

$(\frac{7\pi }{3})*(\frac{180^{\circ}}{\pi }) = 420^{\circ}$

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Example Question #1 : Radians And Conversions

How many radians are in $300^\circ$ ?

Possible Answers:

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

$\pi\sqrt{3}$

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

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

$\frac{7\pi}{3}$

Correct answer:

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

Explanation:

Since $180^\circ=\pi\ \text{radians}$ , we can solve by setting up a proportion:

$\frac{300^\circ}{x}=\frac{180^\circ}{\pi}$

Cross-multiply and solve.

$300^\circ*\pi=180^\circ*x$

$\frac{300^\circ}{180^\circ}\pi=x$

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

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Example Question #1 : Radians And Conversions

How many degrees are in $\frac{5\pi}{6}$ radians?

Possible Answers:

$180^\circ$

$130^\circ$

$210^\circ$

$150^\circ$

$160^\circ$

Correct answer:

$150^\circ$

Explanation:

Since $180^\circ=\pi\ \text{radians}$ , we can solve by setting up a proportion:

$\frac{x}{\frac{5\pi}{6}}=\frac{180^\circ}{\pi}$

Cross multiply and solve.

$180^\circ*\frac{5\pi}{6}=x*\pi$

$\frac{900\pi}{6}=x*\pi$

$150\pi=x*\pi$

$\frac{150\pi}{\pi}=x$

$150^\circ=x$

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Example Question #21 : Unit Circle And Radians

Change $\frac{\pi}{5}$ angle to degrees.

Possible Answers:

$45^{\circ}$

$30^{\circ}$

$32^{\circ}$

$36^{\circ}$

$34^{\circ}$

Correct answer:

$36^{\circ}$

Explanation:

In order to change radians to degrees, we need to multiply the radian agle measure by $\frac{180^{\circ}}{\pi}$ .

$\frac{\pi}{5}\times \frac{180^{\circ}}{\pi}=\frac{180^{\circ}}{5}=36^{\circ}$

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Example Question #11 : Circles

An angle of 40 radians is equal to how many degrees?

Possible Answers:

$\frac{720^{\circ}}{\pi}$

$\frac{5400^{\circ}}{\pi}$

$\frac{3600^{\circ}}{\pi}$

$\frac{7200^{\circ}}{\pi}$

$\frac{7000^{\circ}}{\pi}$

Correct answer:

$\frac{7200^{\circ}}{\pi}$

Explanation:

We know that one radian is equal to $\frac{180^{\circ}}{\pi}$ , so in order to change radians to degrees we need to multiply by $\frac{180^{\circ}}{\pi}$ .

$40\times \frac{180^{\circ}}{\pi}=\frac{7200^{\circ}}{\pi}$

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Example Question #181 : New Sat Math Calculator

Change $\frac{17\pi}{5}$ angle to degrees.

Possible Answers:

$412^{\circ}$

$712^{\circ}$

$512^{\circ}$

$612^{\circ}$

$620^{\circ}$

Correct answer:

$612^{\circ}$

Explanation:

In order to change radians to degrees we need to multiply the radians by $\frac{180^{\circ}}{\pi}$ .

$\frac{17\pi}{5}\times \frac{180^{\circ}}{\pi}=\frac{17\times 180^{\circ}}{5}=612^{\circ}$

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Example Question #182 : New Sat Math Calculator

Change $75^{\circ}$ to radians.

Possible Answers:

$\frac{\pi}{12}$

$\frac{\pi}{2}$

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

$\frac{5\pi}{12}$

$\frac{7\pi}{12}$

Correct answer:

$\frac{5\pi}{12}$

Explanation:

In order to change degrees to radians we need to multiply the degrees by $\frac{\pi}{180^{\circ}}$ .

$75^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{5\pi}{12}$

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Example Question #13 : Radians And Conversions

Change the following expression to degrees:

$\frac{\frac{\pi}{3}-\frac{\pi}{6}}{\frac{1}{3}+\frac{1}{6}}$

Possible Answers:

$60^{\circ}$

$75^{\circ}$

$120^{\circ}$

$90^{\circ}$

$160^{\circ}$

Correct answer:

$60^{\circ}$

Explanation:

First we need to simplify the expression:

$\frac{\frac{\pi}{3}-\frac{\pi}{6}}{\frac{1}{3}+\frac{1}{6}}=\frac{\frac{2\pi-\pi}{6}}{\frac{2+1}{6}}=\frac{\pi}{3}$

Now multiply by $\frac{180^{\circ}}{\pi}$ :

$\frac{\pi}{3}\times \frac{180^{\circ}}{\pi}=\frac{180^{\circ}}{3}=60^{\circ}$

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Example Question #11 : Radians

Change the following expression to degrees:

${\frac{\pi}{4}-\frac{\pi}{12}}$

Possible Answers:

$90^{\circ}$

$60^{\circ}$

$45^{\circ}$

$75^{\circ}$

$30^{\circ}$

Correct answer:

$30^{\circ}$

Explanation:

First we need to simplify the expression:

${\frac{\pi}{4}-\frac{\pi}{12}}=\frac{3\pi-\pi}{12}=\frac{2\pi}{12}=\frac{\pi}{6}$

Then multiply the radians by $\frac{180^{\circ}}{\pi}$ :

$\frac{\pi}{6}\times \frac{180^{\circ}}{\pi}=\frac{180^{\circ}}{6}=30^{\circ}$

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Trigonometry : Unit Circle and Radians

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #1 : Circles

Example Question #1 : Circles

Example Question #1 : Radians And Conversions

Example Question #1 : Radians And Conversions

Example Question #21 : Unit Circle And Radians

Example Question #11 : Circles

Example Question #181 : New Sat Math Calculator

Example Question #182 : New Sat Math Calculator

Example Question #13 : Radians And Conversions

Example Question #11 : Radians

All Trigonometry Resources

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