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Measuring Rotation

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Introduction to Angles of Rotation, Coterminal Angles, and Reference Angles
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In which quadrant does the terminal side of the angle -500^\circ lies and what is the reference angle for this angle?

Guidance

Angles of rotation are formed in the coordinate plane between the positive x -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 and y 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^\circ . The example below illustrates this concept.

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

Example A

Determine two coterminal angles to 837^\circ , one positive and one negative.

Solution: To find coterminal angles we simply add or subtract 360^\circ multiple times to get the angles we desire. 837^\circ - 360^\circ = 477^\circ , so we have a positive coterminal angle. Now we can subtract 360^\circ again to get 477^\circ - 360^\circ=117^\circ .

More Guidance

A reference angle is the acute angle between the terminal side of an angle and the x – 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 – axis. This is a common error for students, especially when the terminal side appears to be closer to the y – axis than the x – axis.

Example B

Determine the quadrant in which -745^\circ lies and hence determine the reference angle.

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

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

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

Example C

Give two coterminal angles to 595^\circ , one positive and one negative, find the reference angle.

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

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^\circ and 235^\circ , so 235^\circ - 180^\circ = 55^\circ .

Concept Problem Revisit Since our angle is more than one rotation, we need to add 360^\circ until we get an angle whose absolute value is less than 360^\circ : -500^\circ + 360^\circ = -200^\circ .

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

Now determine the reference angle: 180^\circ - 160^\circ = 20^\circ .

Guided Practice

1. Find two coterminal angles to 138^\circ , one positive and one negative.

2. Find the reference angle for 895^\circ .

3. Find the reference angle for 343^\circ .

Answers

1. 138^\circ + 360^\circ = 498^\circ and 138^\circ - 360^\circ = -222^\circ

2. 895^\circ - 360^\circ = 535^\circ, 535^\circ - 360^\circ = 175^\circ . The terminal side lies in the second quadrant, so we need to determine the angle between 175^\circ and 180^\circ , which is 5^\circ .

3. 343^\circ is in the fourth quadrant so we need to find the angle between 343^\circ and 360^\circ which is 17^\circ .

Vocabulary

Angle of Rotation
An angle of rotation is formed in the coordinate plane between the positive x -axis (initial side) and a ray (terminal side) . Positive angle measures represent a counterclockwise rotation while negative angles indicate a clockwise rotation.
Coterminal Angle
An angle that has the same terminal and initial sides as another angle, but a different measure.
Reference Angle
The acute angle between the terminal side of an angle and the x –axis.

Practice

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

  1. -98^\circ
  2. 475^\circ
  3. -210^\circ
  4. 47^\circ
  5. -1022^\circ
  6. 354^\circ
  7. -7^\circ

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

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

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