The Unit Circle and Radians - Math

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Question

In the unit circle, what is the angle in radians that corresponds to the point (0, -1)?

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Answer

On the unit circle, (0,-1) is the point that falls between the third and fourth quadrant. This corresponds to $\frac{3\pi }{2}$ .

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Question

What is the reference angle for $\frac{16\pi}{3}$ ?

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Answer

To find the reference angle, subtract $2\pi$ (1 trip around the unit circle) from the given angle until you reach an angle which is less than $2\pi$ .

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Question

Bob manages a pizza store. He bought a new machine that tracks how big his employees are cutting the pizza slices. The machine measures the average angle size of each slice of each pizza. Unfortunately, the angle is given as 0.7854 radians which Bob does not understand. Help Bob by converting the radian angle into degrees. In degrees, what is the size of the angle for an average pizza slice.

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Answer

To convert we use a common conversion amount. It may be easiest to remember the full circle example. In degrees, a full circle is $360^{\circ}$ around. In terms of radians, a full circle is $2\pi$ . So to get our answer

$0.7854\ \text{Radians} *\frac{360^{\circ}}{2\pi\ \text{Radians}}= 45^{\circ}$

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Question

Convert $120^{\circ}$ into radians.

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Answer

To convert from degrees to radians, one multiplies by $\frac{\pi }{180}$ .

$120\cdot \frac{\pi }{180}=\frac{2\pi}{3}$

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Question

What is the value of the angle $\frac{3\pi}{2}$ radians when converted to degrees?

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Answer

The answer can be found using the conversion of 1 radian equals $\frac{180}{\pi}$ degrees. Multiplying $\frac{3\pi}{2}$ by this conversion factor gives 270 degrees.

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Question

Express in radians: $125^{\circ }$

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Answer

Since $180 ^{\circ } = \pi \textrm{ rad}$ , we can convert as follows:

$125 \cdot \frac{\pi }{180} = \frac{25\pi }{36}$

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Question

A point has Cartesian coordinates . Rewrite this as an ordered pair in the polar coordinate plane, rounding the coordinates to the nearest hundredth.

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Answer

Set . Calculate the polar coordinates $(r,\theta )$ as follows:

$r = \sqrt{x^{2}+y^{2}}= \sqrt{1^{2}+3^{2}}=\sqrt{10} \approx 3.16$

$\theta = \arctan \frac{y}{x}= \arctan \frac{3}{1} = \arctan 3 \approx 1.25$

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Question

How many radians are in $270^\circ$ ?

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Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{270^\circ}{x}=\frac{180^\circ}{\pi}$

Cross multiply:

$270^\circ\pi=180^\circx$

Isolate :

$\frac{270^\circ}{180^\circ}\pi=x$

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

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Question

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

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Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

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

Cross multiply:

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

Divide both sides by $\pi$ :

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

$x^\circ=60^\circ$

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Question

How many degrees are in $\frac{8\pi}{3}$ radians?

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Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{x^\circ}{\frac{8\pi}{3}}=\frac{180^\circ}{\pi}$

Cross multiply:

$x^\circ\pi=180^\circ\frac{8\pi}{3}$

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

$x^\circ=180^\circ*\frac{8}{3}$

$x^\circ=480^\circ$

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Question

How many degrees are in $\frac{3\pi}{4}$ radians?

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Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{x^\circ}{\frac{3\pi}{4}}=\frac{180^\circ}{\pi}$

Cross multiply:

$x^\circ\pi=180^\circ\frac{3\pi}{4}$

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

$x^\circ=180^\circ*\frac{3}{4}$

$x^\circ=135^\circ$

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Question

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

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Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

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

Cross multiply:

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

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

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

$x^\circ=120^\circ$

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Question

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

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Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

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

Cross multiply:

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

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

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

$x^\circ=150^\circ$

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Question

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

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Answer

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

Cross multiply:

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

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

$x^\circ=180^\circ*\frac{1}{6}$

$x^\circ=30^\circ$

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Question

How many radians are in $120^\circ$ ?

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Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{120^\circ}{x}=\frac{180^\circ}{\pi}$

Cross multiply:

$120^\circ\pi=180^\circx$

Isolate :

$\frac{120^\circ}{180^\circ}\pi=x$

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

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Question

How many radians are in $360^\circ$ ?

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Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{360^\circ}{x}=\frac{180^\circ}{\pi}$

Cross multiply:

$360^\circ\pi=180^\circx$

Isolate :

$\frac{360^\circ}{180^\circ}\pi=x$

$2\pi=x$

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Question

How many radians are in $405^\circ$ ?

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Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{405^\circ}{x}=\frac{180^\circ}{\pi}$

Cross multiply:

$405^\circ\pi=180^\circx$

Isolate :

$\frac{405^\circ}{180^\circ}\pi=x$

$\frac{9\pi}{4}=x$

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Question

How many radians are in $81^\circ$ ?

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Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{81^\circ}{x}=\frac{180^\circ}{\pi}$

Cross multiply:

$81^\circ\pi=180^\circx$

Isolate :

$\frac{81^\circ}{180^\circ}\pi=x$

Reduce:

$\frac{9\pi}{20}=x$

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Question

How many radians are in $90^\circ$ ?

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Answer

The conversion for radians is $180^\circ=\pi$ , so we can make a ratio:

$\frac{90^\circ}{x}=\frac{180^\circ}{\pi}$

Cross multiply:

$90^\circ\pi=180^\circx$

Isolate :

$\frac{90^\circ}{180^\circ}\pi=x$

Reduce:

$\frac{\pi}{2}=x$

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Question

How many degrees are in $2\pi$ radians?

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Answer

Remember, $180^\circ=\pi$ radians, so $2\pi$ radians would correspond to $2*180^\circ=360^\circ$ .

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