Angle Measures in Degrees and Radians - Pre-Calculus

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Question

Determine the value of $225^\circ$ in radians.

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Answer

To convert from degrees to radians, you do:

$225deg\cdot \frac{(\pi) rad}{180 deg}= \frac{(45\cdot5)\cdot\pi}{45\cdot4} rad= \frac{5\pi}{4} rad$

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Question

Convert $240^{\circ}$ to radians.

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Answer

In order to solve this problem, we must know that

$\text{Radians}=(\frac{\pi }{180^{\circ}})\times \text{Degrees}$

with this formula, we can find our answer:

$(\frac{\pi }{180^{\circ}}) \times 240^{\circ}=\frac{4}{3}\pi$

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Question

Convert $330^{\circ}$ to radians.

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Answer

In order to solve this problem, we need to know that

$\text{Radians}=(\frac{\pi }{180^{\circ}})\times \text{Degrees}$

using this formula, we can find our answer:

$(\frac{\pi }{180^{\circ}})\times330^{\circ}= \frac{11}{6}\pi$

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Question

Convert $\frac{5}{6}\pi$ to degrees.

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Answer

In order to solve this problem, we need to know that

$\text{Degrees}=(\frac{180^{\circ}}{\pi })\times \text{Radians}$

using this formula, we can find our answer:

$(\frac{180^{\circ}}{\pi }) \times \frac{5\pi }{6}= 150^{\circ}$

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Question

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

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Answer

In order to solve this problem, we need to know that

$\text{Degrees}=(\frac{180^{\circ}}{\pi })\times \text{Radians}$

we can arrive to our answer by using this formula:

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

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Question

Convert $180^{\circ}$ to radians.

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Answer

To solve this problem, we need to know that

$\text{Radians}=(\frac{\pi }{180^{\circ}})\times \text{Degrees}$

we can arrive to our answer by using this formula:

$(\frac{\pi }{180^{\circ}})\times 180^{\circ}=\pi$

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Question

Convert the following to degrees:

$\frac{7\pi}{6}$

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Answer

To convert, multiply by the conversion factor $\frac{180}{\pi}$ .

$\frac{7\pi}{6}\frac{180}{\pi}=730=210$

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Question

$\frac{11\pi}{9}$ radians is equivalent to how many degrees?

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Answer

The value of $\pi$ radians equal 180 degrees. Substitute and solve.

$\frac{11\pi}{9}=\frac{11\cdot 180}{9}=\frac{1980}{9}=220$

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Question

Find $90^{\circ}$ expressed as radians.

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Answer

To convert from degrees into radians,

multiply the angle in degrees by $\frac{\pi}{180}$

$90\cdot \frac{\pi}{180}=\frac{\pi}{2}$

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Question

Convert degrees into radians.

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Answer

To convert between degrees and radians, you must use the following conversion factor 180 degrees = pi radians.

Therefore:

$\frac{1140}{1}*\frac{\pi}{180}=\frac{1140\pi}{180}=\frac{19\pi}{3}$

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Question

Please convert the following angle to degrees:

$\frac{7\pi }{6}$

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Answer

To convert from radians to degrees, multiply the input by $\frac{180}{\pi}$ .

In this case:

$\frac{180}{\pi}\cdot \left(\frac{7\pi}{6}\right) = 210^{\circ}$

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Question

Please convert the following from degrees to radians: $315^{\circ}$

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Answer

To convert from degrees to radians, multiply the input by: $\frac{\pi}{180}$

In this case:

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

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Question

Convert $\frac{3\pi}{7}$ radians to degrees. Round to the nearest integer.

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Answer

In order to convert from radians to degrees, we will need to know how many degrees are in a radian.

$\pi \textup{ radians} = 180 \textup{ degrees}$

Substitute 180 as a replacement for $\pi$ .

$\frac{3\pi}{7} = \frac{3\times 180}{7} = \frac{540}{7} = 77.1429$

The correct answer is .

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Question

Convert $50^{\circ}$ to radians.

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Answer

To convert degrees to radians we must recall the conversion factor. Remember that $360^\circ=2\pi$ .

Therefore, we can create the following fraction and solve for the missing variable.

$\frac{50^{\circ}}{360^{\circ}}=\frac{x}{2\pi }$

$360x=2\pi (50)$

$x=\frac{5\pi }{18}$

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Question

Megan, a civil engineer, measures the angle made by the angle of a road. She finds the angle to be $114^{\circ}$ . What is the angle in radians?

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Answer

Megan, a civil engineer, measures the angle made by the angle of a road. She finds the angle to be $114^{\circ}$ . What is the angle in radians?

To convert from degrees to radians, use the following formula:

$\frac{d\pi}{180}=r$

Where and stand for degrees and radians, respectively.

$r=\frac{114\pi}{180}=0.6\bar{3}\pi:\uptext{radians}$

So we get

$0.6\bar{3}\pi :\uptext{radians}$

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Question

While SCUBA diving as part of a search and rescue diving operation, Daniel is responsible for covering a sector which represents $\frac{5\pi}{6}$ radians. What is this angle in degrees?

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Answer

While SCUBA diving as part of a search and rescue diving operation, Daniel is responsible for covering a sector which represents $\frac{5\pi}{6}$ radians. What is this angle in degrees?

Recall the following formula for converting radians to degrees:

$d=\frac{180r}{\pi}$

Where d and r are angle measures in degrees and radians, respectively.

So, we are given a value in radians and asked to find it in degrees. Plug in to the above formula to find the answer!

$d=\frac{180(\frac{5\pi}{6})}{\pi}=150^{\circ}$

So our answer is: $150^{\circ}$

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Question

Terry is running around a circular track. If he covers $\frac{7\pi}{6} radians$ in 120 seconds, how many degrees has he covered?

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Answer

Terry is running around a circular track. If he covers $\frac{7\pi}{6} radians$ in 120 seconds, how many degrees has he covered?

To convert from radians to degrees, use the following helpful formula:

$d=\frac{180r}{\pi}$

Where r and d are our desired angle measure in radians and degrees, respectively.

So, we have r, plug in and simplify to get d!

$d=\frac{180\frac{7\pi}{6} }{\pi}=\frac{30*7\pi}{\pi}=210^{\circ}$

Making our answer:

$210^{\circ}$

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Question

Convert the following to radians.

$270^{\circ}$

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Answer

To solve, use the conversion factor $\frac{\pi{radians}}{180^{\circ}}$ . Thus,

$\frac{270^{\circ}}{1}\cdot \frac{\pi{radians}}{180^{\circ}}=\frac{270\pi}{180}=\frac{3\pi}{2}$

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Question

Convert $660^{\circ}$ to radians.

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Answer

To convert from degrees to radians, you must multiply by $\frac{\pi }{180}$ : $660\cdot \frac{\pi }{180}$ . Then, multiply across as you do normally with fractions. Try and simplify if you can. In this case, 60 goes into both 660 and 180.

Thus, your answer is $\frac{11\pi }{3}$ .

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Question

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

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Answer

To convert from radians to degrees, you must multiply by $\frac{180}{\pi }$ so that you then get $\frac{8\pi }{3}\cdot \frac{180}{\pi }$ . Then, simplify and multiply straight across so that your answer is $480^{\circ}$ .

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