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Trigonometry : Unit Circle and Radians

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Example Questions

Example Question #17 : Circles

Convert the following expression to radians:

$\frac{45^{\circ}+30^{\circ}}{45-30}$

Possible Answers:

$\frac{\pi}{36}$

$\frac{5\pi}{36}$

$\frac{\pi}{12}$

$\frac{\pi}{30}$

$\frac{7\pi}{36}$

Correct answer:

$\frac{\pi}{36}$

Explanation:

First we need to simplify the expression:

$\frac{45^{\circ}+30^{\circ}}{45-30}=\frac{75^{\circ}}{15}=5^{\circ}$

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

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

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Example Question #18 : Circles

Change $270^{\circ}$ to radians.

Possible Answers:

$3\pi$

$\pi$

$2\pi$

$2.5\pi$

$1.5\pi$

Correct answer:

$1.5\pi$

Explanation:

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

$270^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{3\pi}{2}=1.5\pi$

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Example Question #19 : Circles

Convert one radian to degrees.

Possible Answers:

$67.3^{\circ}$

$50^{\circ}$

$60^{\circ}$

$47.3^{\circ}$

$57.3^{\circ}$

Correct answer:

$57.3^{\circ}$

Explanation:

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

$1\times \frac{180^{\circ}}{\pi}=\frac{180^{\circ}}{\pi}\approx 57.3^{\circ}$

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Example Question #191 : New Sat Math Calculator

Convert one degree to radians.

Possible Answers:

$\frac{\pi}{120}$

$\frac{\pi}{90}$

$\pi$

$\frac{\pi}{60}$

$\frac{\pi}{180}$

Correct answer:

$\frac{\pi}{180}$

Explanation:

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

$1^{\circ}\times \frac{\pi}{180^{\circ}}=\frac{\pi}{180}$

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Example Question #21 : Circles

Simplify and give the followoing expression in degrees:

$2\pi-\frac{\pi}{4}$

Possible Answers:

$285^{\circ}$

$270^{\circ}$

$325^{\circ}$

$315^{\circ}$

$300^{\circ}$

Correct answer:

$315^{\circ}$

Explanation:

First we need to simplify the expression:

$2\pi-\frac{\pi}{4}=\frac{8\pi-\pi}{4}=\frac{7\pi}{4}$

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

$\frac{7\pi}{4}\times \frac{180^{\circ}}{\pi}=\frac{7\times 180^{\circ}}{4}=315^{\circ}$

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Example Question #22 : Circles

Simplify and give the followoing expression in degrees:

$\frac{\frac{2\pi}{7}+\frac{\pi}{14}}{\frac{1}{7}}$

Possible Answers:

$450^{\circ}$

$380^{\circ}$

$480^{\circ}$

$420^{\circ}$

$350^{\circ}$

Correct answer:

$450^{\circ}$

Explanation:

First we need to simplify the expression:

$\frac{\frac{2\pi}{7}+\frac{\pi}{14}}{\frac{1}{7}}=\frac{\frac{4\pi+\pi}{14}}{\frac{1}{7}}=\frac{5\pi}{2}$

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

$\frac{5\pi}{2}\times \frac{180^{\circ}}{\pi}=\frac{5\times 180^{\circ}}{2}=450^{\circ}$

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Example Question #23 : Circles

Convert $\frac{2\pi}{3}$ radians to degrees.

Possible Answers:

$120^\circ$

$150^\circ$

$30^\circ$

$250^\circ$

$60^\circ$

Correct answer:

$120^\circ$

Explanation:

Use the conversion $\pi \textup{ radians}= 180\textup{ degrees}$ .

Since we are converting radians to degrees, multiply by 180 degrees and divide by $\pi$ radians.

$\frac{2\pi}{3} \textup{ radians}\left(\frac{180 \textup{ degrees}}{\pi\textup{ radian}}\right)= 120\textup{ degrees}$

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Example Question #192 : New Sat Math Calculator

How many degrees is 3 radians?

Possible Answers:

$150^{\circ}$

$156^{\circ}$

None of these other answers.

$189^{\circ}$

$172^{\circ}$

Correct answer:

$172^{\circ}$

Explanation:

When going from radians to degrees one multiplies by the conversion factor

$\frac{180^{\circ}}{\pi rad}$ .

The radians cancel and the answer is left in degrees.

$3\hspace{1mm}{\color{Red} rad}\cdot\frac{180^{\circ}}{\pi {\color{Red} rad}}=\frac{540^\circ}{\pi}=172^{\circ}$

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Example Question #193 : New Sat Math Calculator

Convert to radians: 33^o

Possible Answers:

0.058

0.189

1.736

0.183

0.576

Correct answer:

0.576

Explanation:

To convert to radians, set up the ratio $\frac{180}{\pi } = \frac{33} { x}$ :

$180 x = 33 \pi$

$x = 33 \pi \div 180 \approx 0.576$

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Example Question #194 : New Sat Math Calculator

Convert to degrees: $\frac{2 \pi }{7}$

Possible Answers:

16.37^o

51.43^o

147.58^o

0.016^o

Correct answer:

51.43^o

Explanation:

To convert, divide by $\pi$ and multiply by 180 :

$\frac{2 \pi }{7 } \div \pi = \frac{2}{7}$

$\frac{2}{7} \cdot 180 \approx 51.47^o$

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Trigonometry : Unit Circle and Radians

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #17 : Circles

Example Question #18 : Circles

Example Question #19 : Circles

Example Question #191 : New Sat Math Calculator

Example Question #21 : Circles

Example Question #22 : Circles

Example Question #23 : Circles

Example Question #192 : New Sat Math Calculator

Example Question #193 : New Sat Math Calculator

Example Question #194 : New Sat Math Calculator

All Trigonometry Resources

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