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Trigonometry : Unit Circle and Radians

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Example Question #42 : Radians And Conversions

Convert $\frac{11\pi}{5}$ into degrees.

Possible Answers:

$400\degree$

$300\degree$

$396\degree$

$395\degree$

Correct answer:

$396\degree$

Explanation:

To convert from radians to degrees, you must multiply the angle by $\frac{180\degree}{\pi}$ to cancel the $\pi$ units.

This will get you

$11\cdot \frac{180\degree}{5}=396^\circ$

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Example Question #212 : New Sat Math Calculator

Convert $\frac{2\pi}{5}$ to degrees:

Possible Answers:

$72^{\circ}$

$36^{\circ}$

$72\pi$

$\frac{2}{5}^{\circ}$

$114^{\circ}$

Correct answer:

$72^{\circ}$

Explanation:

To convert radians to degrees, we need to multiply the given radians by $\frac{180^{\circ}}{\pi}$ .

$\frac{2\pi}{5}*\frac{180^{\circ}}{\pi}=\frac{360^{\circ}}{5}=72^{\circ}$

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Example Question #41 : Radians

Give your answer in terms of $\pi$ .

Convert $165^{\circ}$ to radians:

Possible Answers:

$\frac{11\pi}{24}$

$\frac{11\pi}{12}$

$\frac{12\pi}{11}$

$\frac{\pi}{12}$

$\frac{11\pi}{6}$

Correct answer:

$\frac{11\pi}{12}$

Explanation:

To convert degrees to radians, we need to multiply the given degree by $\frac{\pi}{180^{\circ}}$ .

$165*^{\circ}\frac{\pi}{180^{\circ}}=\frac{165\pi}{180}$

To simplify, we get:

$\frac{11\pi}{12}$

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Example Question #61 : Unit Circle And Radians

Convert $52^{\circ}$ to radians.

Possible Answers:

$\tfrac{13}{45}\pi$

$\frac{\pi}{2}$

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

$\tfrac{17}{12}\pi$

$\frac{\pi}{5}$

Correct answer:

$\tfrac{13}{45}\pi$

Explanation:

To convert between degrees and radians, multiply by $\frac{\pi }{180^{\circ}}$ or $\frac{180^{\circ}}{\pi}$ .

Since we start in degrees, we use $\frac{\pi }{180^{\circ}}$ :

$\frac{52^{\circ}}{1}\times\frac{\pi}{180^{\circ}}=\frac{52}{180}\pi$

which simplifies to $\tfrac{13}{45}\pi$ .

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Example Question #51 : Radians And Conversions

Convert $\frac{7\pi}{4}$ radians into degrees.

Possible Answers:

$285^{\circ}$

$345^{\circ}$

$270^{\circ}$

$225^{\circ}$

$315^{\circ}$

Correct answer:

$315^{\circ}$

Explanation:

Recall the definition of "radians" derived from the unit circle:

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

The quantity of radians given in the problem is $\frac{7\pi}{4}$ . All that is required to convert this measure into degrees is to denote the unknown angle measure in degrees by $\Theta$ and set up a proportion equation using the aforementioned definition relating radians to degrees:

$\frac{180^{\circ}}{\Theta} = \frac{\pi rad}{\frac{7\pi}{4} rad}$

Cross-multiply the denominators in these fractions to obtain:

$1260^{\circ}\pi rad=4\Theta\pi rad$

$315^{\circ}\pi rad =\Theta\pi rad$ .

Canceling like terms in these equations yields

$\Theta = 315^{\circ}$

Hence, the correct angle measure of $\frac{7\pi}{4}$ in degrees is $315^{\circ}$ .

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Example Question #1 : Angles In The Unit Circle

What are the ways to write 360^o and 720^o in radians?

Possible Answers:

$\small \small 0,\pi$

$\small 2\pi,4\pi$

$\small \frac{\pi}{4},\frac{\pi}{2}$

$\small \pi,2\pi$

$\small \frac{\pi}{2},\pi$

Correct answer:

$\small 2\pi,4\pi$

Explanation:

$\small 360^o=2\pi$ on the unit circle.

$\small \small \small 720^o=4\pi$ on the unit circle.

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Example Question #1 : Angles In The Unit Circle

Which of the following is not an angle in the unit circle?

Possible Answers:

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

$\frac{\pi}{5}$

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

$\pi$

$\frac{5\pi}{6}$

Correct answer:

$\frac{\pi}{5}$

Explanation:

The unit circle is in two increments: $\frac{\pi}{6}$ : $(\frac{\pi}{6},\frac{2\pi}{6},\frac{3\pi}{6})$ , etc. and $\frac{\pi}{4}$ : $(\frac{\pi}{4},\frac{2\pi}{4},\frac{3\pi}{4})$ , etc. The only answer choice that is not a multiple of either $\frac{\pi}{6}$ or $\frac{\pi}{4}$ is $\frac{\pi}{5}$ .

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Example Question #1 : Angles In The Unit Circle

What is the equivalent of $72^{\circ}$ , in radians?

Possible Answers:

$\frac{2\pi}{10}$

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

$\frac{\pi }{10}$

$\frac{\pi }{5}$

$\pi$

Correct answer:

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

Explanation:

To convert from degrees to radians, use the equality

$2\pi = 360^{\circ}$ or $\pi = 180^{\circ}$ .

From here we use $\frac{\pi}{180^{\circ}}$ as a unit multiplier to convert our degrees into radians.

$\left (72^{\circ} \right )\left ( \frac{\pi }{180^{\circ}} \right ) = \frac{2\pi}{5}$

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Example Question #1 : Angles In The Unit Circle

What is the value of tan(45) ?

Possible Answers:

$\frac{1}{2}$

$\frac{\sqrt{2}}{2}$

$\frac{\sqrt{3}}{2}$

Correct answer:

Explanation:

To help with this one, draw a 45-45-90 triangle. With legs equal to 1 and a hypotenuse of $\sqrt2$ .

Then, use the definition of tangent as opposite over adjacent to find the value.

Since the legs are congruent, we get that the ratio is 1.

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Example Question #2 : Angles In The Unit Circle

What is sin(90) ?

Possible Answers:

$\frac{1}{2}$

$\frac{\sqrt{3}}{2}$

Correct answer:

Explanation:

Recall that on the unit circle, sine represents the y coordinate of the unit circle.

Then, since we are at 90˚, we are at the positive y axis, the point (0,1).

At this point on the unit circle, the y value is 1.

Thus sin(90)=1 .

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Trigonometry : Unit Circle and Radians

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #42 : Radians And Conversions

Example Question #212 : New Sat Math Calculator

Example Question #41 : Radians

Example Question #61 : Unit Circle And Radians

Example Question #51 : Radians And Conversions

Example Question #1 : Angles In The Unit Circle

Example Question #1 : Angles In The Unit Circle

Example Question #1 : Angles In The Unit Circle

Example Question #1 : Angles In The Unit Circle

Example Question #2 : Angles In The Unit Circle

All Trigonometry Resources

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