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### Unit Circle and Radian Measure

The unit circle is the circle centered at the origin with radius equal to one unit. This means that the distance from the origin to any point on the circle is equal to one unit.

Using the unit circle, we can define another unit of measure for angles, radians. Radian measure is based upon the circumference of the unit circle. The circumference of the unit circle is

One radian is equal to the measure of

We can use the equality,

To convert from degrees to radians, multiply by

To convert from radians to degrees, multiply by

Let's convert the following units of measure.

- Convert
250∘ to radians.

To convert from degrees to radians, multiply by

- Convert
3π to degrees.

To convert from radians to degrees, multiply by

Now, let's find two angles, one positive and one negative, coterminal to

Since we are working in radians now we will add/subtract multiple of

Now, to find the reference angle, first determine in which quadrant

Consider

Finally, let's find two angles coterminal to

This time we will add multiples of

In this case

### Examples

#### Example 1

Earlier, you were asked to convert

To convert from degrees to radians, multiply by

**Convert the following angle measures from degrees to radians.**

#### Example 2

#### Example 3

#### Example 4

\begin{align*}330^\circ\end{align*}

\begin{align*}330^\circ \times \frac{\pi}{180^\circ}=\frac{11 \pi}{6}\end{align*}

**Convert the following angle measures from radians to degrees.**

#### Example 5

\begin{align*}\frac{5 \pi}{6}\end{align*}

\begin{align*}\frac{5 \pi}{6} \times \frac{180^\circ}{\pi}=150^\circ\end{align*}

#### Example 6

\begin{align*}\frac{13 \pi}{4}\end{align*}

\begin{align*}\frac{13 \pi}{4} \times \frac{180^\circ}{\pi}=585^\circ\end{align*}

#### Example 7

\begin{align*}-\frac{5 \pi}{2}\end{align*}

\begin{align*}-\frac{5 \pi}{2} \times \frac{180^\circ}{\pi}=-450^\circ\end{align*}

#### Example 8

Find two coterminal angles to \begin{align*}\frac{11 \pi}{4}\end{align*}, one positive and one negative, and its reference angle.

There are many possible coterminal angles, here are some possibilities:

positive coterminal angle: \begin{align*}\frac{11 \pi}{4} + \frac{8 \pi}{4} = \frac{19 \pi}{4}\end{align*} or \begin{align*}\frac{11 \pi}{4} - \frac{8 \pi}{4} = \frac{3 \pi}{4}\end{align*},

negative coterminal angle: \begin{align*}\frac{11 \pi}{4} - \frac{16 \pi}{4} = -\frac{5 \pi}{4}\end{align*} or \begin{align*}\frac{11 \pi}{4} - \frac{24 \pi}{4} = -\frac{13 \pi}{4}\end{align*}

Using the coterminal angle, \begin{align*}\frac{3 \pi}{4}\end{align*}, which is \begin{align*}\frac{\pi}{4}\end{align*} from \begin{align*}\frac{4 \pi}{4}\end{align*}. So the terminal side lies in the second quadrant and the reference angle is \begin{align*}\frac{\pi}{4}\end{align*}.

### Review

For problems 1-5, convert the angle from degrees to radians. Leave answers in terms of \begin{align*}\pi\end{align*}.

- \begin{align*}135^\circ\end{align*}
- \begin{align*}240^\circ\end{align*}
- \begin{align*}-330^\circ\end{align*}
- \begin{align*}450^\circ\end{align*}
- \begin{align*}-315^\circ\end{align*}

For problems 6-10, convert the angle measure from radians to degrees.

- \begin{align*}\frac{7 \pi}{3}\end{align*}
- \begin{align*}-\frac{13 \pi}{6}\end{align*}
- \begin{align*}\frac{9 \pi}{2}\end{align*}
- \begin{align*}-\frac{3 \pi}{4}\end{align*}
- \begin{align*}\frac{5 \pi}{6}\end{align*}

For problems 11-15, find two coterminal angles (one positive, one negative) and the reference angle for each angle in radians.

- \begin{align*}\frac{8 \pi}{3}\end{align*}
- \begin{align*}\frac{11 \pi}{4}\end{align*}
- \begin{align*}-\frac{\pi}{6}\end{align*}
- \begin{align*}\frac{4 \pi}{3}\end{align*}
- \begin{align*}-\frac{17 \pi}{6}\end{align*}

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 13.6.