1.5: Measuring Rotation
Learning Objectives
 Identify and draw angles of rotation in standard position.
 Identify quadrantal angles.
 Identify coterminal angles.
Angles of Rotation in Standard Position
Consider, for example, a game that is played with a spinner. When you spin the spinner, how far has it gone? You can answer this question in several ways. You could say something like “the spinner spun around 3 times.” This means that the spinner made 3 complete rotations, and then landed back where it started.
We can also measure the rotation in degrees. In the previous lesson we worked with angles in triangles, measured in degrees. You may recall from geometry that a full rotation is 360 degrees, usually written as
We can use our knowledge of graphing to represent any angle. The figure below shows an angle in what is called standard position.
The initial side of an angle in standard position is always on the positive
The 90 degree angle is one of four quadrantal angles. A quadrantal angle is one whose terminal side lies on an axis. Along with
These angles are referred to as quadrantal because each angle defines a quadrant. Notice that without the arrow indicating the rotation,
Coterminal Angles
Consider the angle
Now consider the angle
Notice that
Example 1: Which angles are coterminal with
a.
b.
c.
d.
Solution: b.
Notice that terminal side of the first angle,
Now consider
Points to Consider
 How can one angle look exactly the same as another angle?
 Where might you see angles of rotation in real life?
Review Questions
 Plot the following angles in standard position.

60∘ 
−170∘ 
365∘ 
325∘ 
240∘

 State the measure of an angle that is coterminal with
90∘.  Name a positive and negative angle that are coterminal with:

120∘ 
315∘ 
−150∘

 A drag racer goes around a 180 degree circular curve in a racetrack in a path of radius 120 m. Its front and back wheels have different diameters. The front wheels are 0.6 m in diameter. The rear wheels are much larger; they have a diameter of 1.8 m. The axles of both wheels are 2 m long. Which wheel has more rotations going around the curve? How many more degrees does the front wheel rotate compared to the back wheel?
Review Answers

 Answers will vary. Examples:
450∘, −270∘  Answers will vary. Examples:
−240∘, 480∘  Answers will vary. Examples:
−45∘, 675∘  Answers will vary. Examples: \begin{align*}210^\circ, \ 510^\circ, \ 570^\circ\end{align*}
210∘, −510∘, 570∘
 Answers will vary. Examples:
 The front wheel rotates more because it has a smaller diameter. It rotates 200 revolutions versus 66.67 revolutions for the back wheel, which is a \begin{align*}48,000^\circ\end{align*}
48,000∘ difference \begin{align*}((20066.\bar{6}) \cdot 360^\circ)\end{align*}((200−66.6¯)⋅360∘) .
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