3. Trigonometry

a. Angular Measure

Degrees

Most people are used to using degrees to measure angles. There are \(360^\circ\) in a full circle, \(90^\circ\) in a right angle and \(180^\circ\) in a straight angle.

Radians

However, mathematicians and most scientists and engineers use radian measure. There are \(2\pi\) rad in a full circle, \(\dfrac{\pi}{2}\) rad in a right angle and \(\pi\) rad in a straight angle. You need to be able to go back and forth between radians and degrees.

Dr. Y's Robot

The following activity may help you to conceptualize radians.
• Stand up and face the ccreen. Make Dr. Y's robot follow the commands and do them yourself.
• Raise or lower the robot's arms by draging the hand or clicking on the up or down arrows.
• Rotate the robot by dragging the dot or clicking on the counterclockwise or clockwise arrows.
• Reset the robot by clicking on the reset button.
• Get a new problem by clicking on the refresh button.

  1. Put out your left arm. Turn toward the left until you are facing backwards. This is called turning by \(\pi\) rad.  That's    \(\pi\)
    2. Go back to your starting position.
  2. Put out your left arm. Turn toward the left until you are facing left. That is called turning by \(\dfrac{1}{2}\pi\) rad.  That's    \(\pi\)
    3. Go back to your starting position.
  3. Turn by \(\dfrac{1}{4}\pi\) rad.  That's    \(\pi\)

The coordinates on a number line are positive on the right and negative on the left. The angles to measure rotations also have signs. Turning to the left is positive as above. Turning to the right is negative as follows:

    Go back to your starting position.
  1. Put out your right arm. Turn toward the right until you are facing right. That is called turning by \(-\,\dfrac{\pi}{2}\) rad.  That's    \(\pi\)
    2. Go back to your starting position.
  2. Put out your right arm. Turn toward the right until you are facing backwards. That is called turning by \(-\pi\) rad.  That's    \(\pi\)
    3. Go back to your starting position.
  3. Turn by \(-\,\dfrac{3}{4}\pi\) rad.  That's    \(\pi\)

Directions in the \(xy\)-plane are also specified by angular measure that may be positive or negative. The positive \(x\)-axis is called \(0^\circ\) or \(0\) rad. Directions which are counterclockwise (turning left) from the positive \(x\)-axis are positive. Directions which are clockwise (turning right) from the positive \(x\)-axis are negative.

    Go back to your starting position.
  • Put out your left arm. Turn counterclockwise until you are facing left. That is the \(\dfrac{\pi}{2}\) rad direction.  That's    \(\pi\)
    • Go back to your starting position.
  • Put out your right arm. Turn clockwise until you are facing right. That is the \(-\,\dfrac{\pi}{2}\) rad direction.  That's    \(\pi\)
    • Go back to your starting position.
  • Turn to the \(\dfrac{2}{3}\pi\) rad direction.  That's    \(\pi\)
  • Start at the \(\dfrac{2}{3}\pi\) rad direction. Then turn by \(\dfrac{2}{3}\pi\) rad.  That's    \(\pi\)

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Supported in part by NSF Grant #1123255