The Unit Circle and Radians - Math

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

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

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

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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 $2\pi$ .

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

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

Convert into radians.

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

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

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

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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.

Express in radians:

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

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

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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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$

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

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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}$=x$

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

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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}$$

$x^\circ=60^\circ$

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

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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}$$

$x^\circ=480^\circ$

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

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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}$$

$x^\circ=135^\circ$

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

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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}$$

$x^\circ=120^\circ$

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

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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}$$

$x^\circ=150^\circ$

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

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

$x^\circ=30^\circ$

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

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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}$=x$

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

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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}$\pi=x$

$2\pi=x$

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

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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}$=x$

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

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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}$=x$

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

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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}$=x$

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

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