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# Measuring Rotation

## Explore trig ratios of angles greater than 90 degrees

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Introduction to Angles of Rotation, Coterminal Angles, and Reference Angles

In which quadrant does the terminal side of the angle 500\begin{align*}-500^\circ\end{align*} lies and what is the reference angle for this angle?

### Angles of Rotation

Angles of rotation are formed in the coordinate plane between the positive x\begin{align*}x\end{align*}-axis (initial side) and a ray (terminal side). Positive angle measures represent a counterclockwise rotation while negative angles indicate a clockwise rotation.

Since the x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} axes are perpendicular, each axis then represents an increment of ninety degrees of rotation. The diagrams below show a variety of angles formed by rotating a ray through the quadrants of the coordinate plane.

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 360\begin{align*}360^\circ\end{align*}. The example below illustrates this concept.

For the angle 525\begin{align*}525^\circ\end{align*}, an entire 360\begin{align*}360^\circ\end{align*} rotation is made and then we keep going another 165\begin{align*}165^\circ\end{align*} to 525\begin{align*}525^\circ\end{align*}. Therefore, the resulting angle is equivalent to 525360\begin{align*}525^\circ - 360^\circ\end{align*}, or 165\begin{align*}165^\circ\end{align*}. In other words, the terminal side is in the same location as the terminal side for a 165\begin{align*}165^\circ\end{align*} angle. If we subtract 360\begin{align*}360^\circ\end{align*} again, we get a negative angle, 195\begin{align*}-195^\circ\end{align*}. Since they all share the same terminal side, they are called coterminal angles.

#### Solve the following problem

Determine two coterminal angles to 837\begin{align*}837^\circ\end{align*}, one positive and one negative.

To find coterminal angles we simply add or subtract 360\begin{align*}360^\circ\end{align*} multiple times to get the angles we desire. 837360=477\begin{align*}837^\circ - 360^\circ = 477^\circ\end{align*}, so we have a positive coterminal angle. Now we can subtract 360\begin{align*}360^\circ\end{align*} again to get 477360=117\begin{align*}477^\circ - 360^\circ=117^\circ\end{align*}.

### Reference Angle

A reference angle is the acute angle between the terminal side of an angle and the x\begin{align*}x\end{align*} – axis. The diagram below shows the reference angles for terminal sides of angles in each of the four quadrants.

Note: A reference angle is never determined by the angle between the terminal side and the y\begin{align*}y\end{align*} – axis. This is a common error for students, especially when the terminal side appears to be closer to the y\begin{align*}y\end{align*} – axis than the x\begin{align*}x\end{align*} – axis.

#### Solve the following problems

Determine the quadrant in which 745\begin{align*}-745^\circ\end{align*} lies and hence determine the reference angle.

Since our angle is more than one rotation, we need to add 360\begin{align*}360^\circ\end{align*} until we get an angle whose absolute value is less than 360\begin{align*}360^\circ\end{align*}: 745+360=385\begin{align*}-745^\circ + 360^\circ = -385^\circ\end{align*}, again 385+360=25\begin{align*}-385^\circ + 360^\circ = -25^\circ\end{align*}.

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

Note that the reference angle is positive 25\begin{align*}25^\circ\end{align*}. All reference angles will be positive as they are acute angles (between 0\begin{align*}0^\circ\end{align*} and 90\begin{align*}90^\circ\end{align*}).

Give two coterminal angles to 595\begin{align*}595^\circ\end{align*}, one positive and one negative, find the reference angle.

To find the coterminal angles we can add/subtract 360\begin{align*}360^\circ\end{align*}. In this case, our angle is greater than 360\begin{align*}360^\circ\end{align*} so it makes sense to subtract 360\begin{align*}360^\circ\end{align*} to get a positive coterminal angle: 595360=235\begin{align*}595^\circ - 360^\circ = 235^\circ\end{align*}. Now subtract again to get a negative angle: 235360=125\begin{align*}235^\circ - 360^\circ = -125^\circ\end{align*}.

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 180\begin{align*}180^\circ\end{align*} and 235\begin{align*}235^\circ\end{align*}, so 235180=55\begin{align*}235^\circ - 180^\circ = 55^\circ\end{align*}.

### Examples

#### Example 1

Earlier, you were asked what is the reference angle.

Since our angle is more than one rotation, we need to add 360\begin{align*}360^\circ\end{align*} until we get an angle whose absolute value is less than 360\begin{align*}360^\circ\end{align*}: 500+360=200\begin{align*}-500^\circ + 360^\circ = -200^\circ\end{align*}.

If we plot this angle we see that it is 200\begin{align*}-200^\circ\end{align*} clockwise from the origin or 160\begin{align*}160^\circ\end{align*} counterclockwise. 160\begin{align*}160^\circ\end{align*} lies in the second quadrant.

Now determine the reference angle: 180160=20\begin{align*}180^\circ - 160^\circ = 20^\circ\end{align*}.

#### Example 2

Find two coterminal angles to 138\begin{align*}138^\circ\end{align*}, one positive and one negative.

138+360=498\begin{align*}138^\circ + 360^\circ = 498^\circ\end{align*} and 138360=222\begin{align*}138^\circ - 360^\circ = -222^\circ\end{align*}

#### Example 3

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

895360=535,535360=175\begin{align*}895^\circ - 360^\circ = 535^\circ, 535^\circ - 360^\circ = 175^\circ\end{align*}. The terminal side lies in the second quadrant, so we need to determine the angle between 175\begin{align*}175^\circ\end{align*} and 180\begin{align*}180^\circ\end{align*}, which is 5\begin{align*}5^\circ\end{align*}.

#### Example 4

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

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

### Review

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

1. 98\begin{align*}-98^\circ\end{align*}
2. 475\begin{align*}475^\circ\end{align*}
3. 210\begin{align*}-210^\circ\end{align*}
4. 47\begin{align*}47^\circ\end{align*}
5. 1022\begin{align*}-1022^\circ\end{align*}
6. 354\begin{align*}354^\circ\end{align*}
7. 7\begin{align*}-7^\circ\end{align*}

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

1. 102\begin{align*}102^\circ\end{align*}
2. 400\begin{align*}-400^\circ\end{align*}
3. 1307\begin{align*}1307^\circ\end{align*}
4. 820\begin{align*}-820^\circ\end{align*}
5. 304\begin{align*}304^\circ\end{align*}
6. 251\begin{align*}251^\circ\end{align*}
7. 348\begin{align*}-348^\circ\end{align*}
8. Explain why the reference angle for an angle between 0\begin{align*}0^\circ\end{align*} and 90\begin{align*}90^\circ\end{align*} is equal to itself.

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

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