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1.5: Measuring Rotation

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

  • Identify and draw angles of rotation in standard position.
  • Identify quadrantal angles.
  • Identify co-terminal 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 \begin{align*}360^\circ\end{align*}360. Half a rotation is then \begin{align*}180^\circ\end{align*}180 and a quarter rotation is \begin{align*}90^\circ\end{align*}90. Each of these measurements will be important in this lesson, as well as in the remainder of the chapter.

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 \begin{align*}x-\end{align*}xaxis. The terminal side always meets the initial side at the origin. Notice that the rotation goes in a counterclockwise direction. This means that if we rotate clockwise, we will generate a negative angle. Below are several examples of angles in standard position.

The 90 degree angle is one of four quadrantal angles. A quadrantal angle is one whose terminal side lies on an axis. Along with \begin{align*}90^\circ, \ 0^\circ, \ 180^\circ\end{align*}90, 0, 180 and \begin{align*}270^\circ\end{align*}270 are quadrantal angles.

These angles are referred to as quadrantal because each angle defines a quadrant. Notice that without the arrow indicating the rotation, \begin{align*}270^\circ\end{align*}270 looks as if it is a \begin{align*}-90^\circ\end{align*}90, defining the fourth quadrant. Notice also that \begin{align*}360^\circ\end{align*}360 would look just like \begin{align*}0^\circ\end{align*}0. The difference is in the action of rotation. This idea of two angles actually being the same angle is discussed next.

Coterminal Angles

Consider the angle \begin{align*}30^\circ\end{align*}30, in standard position.

Now consider the angle \begin{align*}390^\circ\end{align*}390. We can think of this angle as a full rotation \begin{align*}(360^\circ)\end{align*}(360), plus an additional 30 degrees.

Notice that \begin{align*}390^\circ\end{align*}390 looks the same as \begin{align*}30^\circ\end{align*}30. Formally, we say that the angles share the same terminal side. Therefore we call the angles co-terminal. Not only are these two angles co-terminal, but there are infinitely many angles that are co-terminal with these two angles. For example, if we rotate another \begin{align*}360^\circ\end{align*}360, we get the angle \begin{align*}750^\circ\end{align*}750. Or, if we create the angle in the negative direction (clockwise), we get the angle \begin{align*}-330^\circ\end{align*}330. Because we can rotate in either direction, and we can rotate as many times as we want, we can continuously generate angles that are co-terminal with \begin{align*}30^\circ\end{align*}30.

Example 1: Which angles are co-terminal with \begin{align*}45^\circ\end{align*}45?

a. \begin{align*}-45^\circ\end{align*}45

b. \begin{align*}405^\circ\end{align*}405

c. \begin{align*}-315^\circ\end{align*}

d. \begin{align*}135^\circ\end{align*}

Solution: b. \begin{align*}405^\circ\end{align*} and c. \begin{align*}-315^\circ\end{align*} are co-terminal with \begin{align*}45^\circ\end{align*}.

Notice that terminal side of the first angle, \begin{align*}-45^\circ\end{align*}, is in the \begin{align*}4^{th}\end{align*} quadrant. The last angle, \begin{align*}135^\circ\end{align*} is in the \begin{align*}2^{nd}\end{align*} quadrant. Therefore neither angle is co-terminal with \begin{align*}45^\circ\end{align*}.

Now consider \begin{align*}405^\circ\end{align*}. This is a full rotation, plus an additional 45 degrees. So this angle is co-terminal with \begin{align*}45^\circ\end{align*}. The angle \begin{align*}-315^\circ\end{align*} can be generated by rotating clockwise. To determine where the terminal side is, it can be helpful to use quadrantal angles as markers. For example, if you rotate clockwise 90 degrees 3 times (for a total of 270 degrees), the terminal side of the angle is on the positive \begin{align*}y-\end{align*}axis. For a total clockwise rotation of 315 degrees, we have \begin{align*}315-270 = 45\end{align*} degrees more to rotate. This puts the terminal side of the angle at the same position as \begin{align*}45^\circ\end{align*}.

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

  1. Plot the following angles in standard position.
    1. \begin{align*}60^\circ\end{align*}
    2. \begin{align*}-170^\circ\end{align*}
    3. \begin{align*}365^\circ\end{align*}
    4. \begin{align*}325^\circ\end{align*}
    5. \begin{align*}240^\circ\end{align*}
  2. State the measure of an angle that is co-terminal with \begin{align*}90^\circ.\end{align*}
  3. Name a positive and negative angle that are co-terminal with:
    1. \begin{align*}120^\circ\end{align*}
    2. \begin{align*}315^\circ\end{align*}
    3. \begin{align*}-150^\circ\end{align*}
  4. 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

  1. Answers will vary. Examples: \begin{align*}450^\circ, \ -270^\circ\end{align*}
    1. Answers will vary. Examples: \begin{align*}-240^\circ, \ 480^\circ\end{align*}
    2. Answers will vary. Examples: \begin{align*}-45^\circ, \ 675^\circ\end{align*}
    3. Answers will vary. Examples: \begin{align*}210^\circ, \ -510^\circ, \ 570^\circ\end{align*}
  2. 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*} difference \begin{align*}((200-66.\bar{6}) \cdot 360^\circ)\end{align*}.

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