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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 500o\begin{align*}-500^{\text{o}}\end{align*} lie 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 360o\begin{align*}360^{\text{o}}\end{align*}. The example below illustrates this concept.

For the angle 525o\begin{align*}525^{\text{o}}\end{align*}, an entire 360o\begin{align*}360^{\text{o}}\end{align*} rotation is made and then we keep going another 165o\begin{align*}165^{\text{o}}\end{align*} to 525o\begin{align*}525^{\text{o}}\end{align*}. Therefore, the resulting angle is equivalent to 525o360o\begin{align*}525^\text{o} - 360^\text{o}\end{align*}, or 165o\begin{align*}165^\text{o}\end{align*}. In other words, the terminal side is in the same location as the terminal side for a 165o\begin{align*}165^\text{o}\end{align*} angle. If we subtract 360o\begin{align*}360^\text{o}\end{align*} again, we get a negative angle, 195o\begin{align*}-195^\text{o}\end{align*}. Since they all share the same terminal side, they are called coterminal angles.

Let's determine two coterminal angles to 837o\begin{align*}837^\text{o}\end{align*}, one positive and one negative.

To find coterminal angles we simply add or subtract 360o\begin{align*}360^\text{o}\end{align*} multiple times to get the angles we desire. 837o360o=477o\begin{align*}837^\text{o} - 360^\text{o} = 477^\text{o}\end{align*}, so we have a positive coterminal angle. Now we can subtract 360o\begin{align*}360^\text{o}\end{align*} again to get 477o360o=117o\begin{align*}477^\text{o} - 360^\text{o}=117^\text{o}\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.

Now, let's determine the quadrant in which 745o\begin{align*}-745^\text{o}\end{align*} lies and hence determine the reference angle.

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

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

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

Finally, let's give two coterminal angles to 595\begin{align*}595^\circ\end{align*}, one positive and one negative, and 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 to find the reference angle of 500o\begin{align*}-500^\text{o}\end{align*} and find the quadrant in which the terminal side lies.

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

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

Now determine the reference angle: 180o160o=20o\begin{align*}180^\text{o} - 160^\text{o} = 20^\text{o}\end{align*}.

#### Example 2

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

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

#### 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.

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

1. \begin{align*}102^\text{o}\end{align*}
2. \begin{align*}-400^\text{o}\end{align*}
3. \begin{align*}1307^\text{o}\end{align*}
4. \begin{align*}-820^\text{o}\end{align*}
5. \begin{align*}304^\text{o}\end{align*}
6. \begin{align*}251^\text{o}\end{align*}
7. \begin{align*}-348^\text{o}\end{align*}
8. 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.

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

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