In which quadrant does the terminal side of the angle

### Angles of Rotation

Angles of rotation are formed in the coordinate plane between the positive **(initial side)** and a ray **(terminal side)**. Positive angle measures represent a counterclockwise rotation while negative angles indicate a clockwise rotation.

Since the

An angle of rotation can be described infinitely many ways. It can be described by a positive or negative angle of rotation or by making multiple full circle rotations through

For the angle **coterminal angles**.

Let's determine two coterminal angles to

To find coterminal angles we simply add or subtract

### Reference Angle

A **reference angle** is the acute angle between the terminal side of an angle and the

Note: A reference angle is never determined by the angle between the terminal side and the

Now, let's determine the quadrant in which

Since our angle is more than one rotation, we need to add

Now we can plot the angle and determine the reference angle:

Note that the reference angle is positive

Finally, let's give two coterminal angles to

To find the coterminal angles we can add/subtract

By plotting any of these angles we can see that the terminal side lies in the third quadrant as shown.

Since the terminal side lies in the third quadrant, we need to find the angle between

### Examples

#### Example 1

Earlier, you were asked to find the reference angle of

Since our angle is more than one rotation, we need to add

If we plot this angle we see that it is

Now determine the reference angle:

#### Example 2

Find two coterminal angles to

#### Example 3

Find the reference angle for \begin{align*}895^\text{o}\end{align*}.

\begin{align*}895^\text{o} - 360^\text{o} = 535^\text{o}, 535^\text{o} - 360^\text{o} = 175^\text{o}\end{align*}. The terminal side lies in the second quadrant, so we need to determine the angle between \begin{align*}175^\text{o}\end{align*} and \begin{align*}180^\text{o}\end{align*}, which is \begin{align*}5^\text{o}\end{align*}.

#### Example 4

Find the reference angle for \begin{align*}343^\text{o}\end{align*}.

\begin{align*}343^\text{o}\end{align*} is in the fourth quadrant so we need to find the angle between \begin{align*}343^\text{o}\end{align*} and \begin{align*}360^\text{o}\end{align*} which is \begin{align*}17^\text{o}\end{align*}.

### Review

Find two coterminal angles to each angle measure, one positive and one negative.

- \begin{align*}-98^\text{o}\end{align*}
- \begin{align*}475^\text{o}\end{align*}
- \begin{align*}-210^\text{o}\end{align*}
- \begin{align*}47^\text{o}\end{align*}
- \begin{align*}-1022^\text{o}\end{align*}
- \begin{align*}354^\text{o}\end{align*}
- \begin{align*}-7^\text{o}\end{align*}

Determine the quadrant in which the terminal side lies and find the reference angle for each of the following angles.

- \begin{align*}102^\text{o}\end{align*}
- \begin{align*}-400^\text{o}\end{align*}
- \begin{align*}1307^\text{o}\end{align*}
- \begin{align*}-820^\text{o}\end{align*}
- \begin{align*}304^\text{o}\end{align*}
- \begin{align*}251^\text{o}\end{align*}
- \begin{align*}-348^\text{o}\end{align*}
- Explain why the reference angle for an angle between \begin{align*}0^\text{o}\end{align*} and \begin{align*}90^\text{o}\end{align*} is equal to itself.

### Answers for Review Problems

To see the Review answers, open this PDF file and look for section 13.5.