Compare Your Answers
  1. Determine the reference angle for each angle in standard position.

    1. \(166^\circ\)

      Because this angle is less than \(180^\circ\) and more than \(90^\circ\) , it terminates in Quadrant II.

      The reference angle is:

      \(\begin{align}
       \theta _R &= 180^\circ - \theta  \\
       \theta _R &= 180^\circ - 166^\circ  \\
       \theta _R &= 14^\circ  \\
       \end{align}\)

    2. \(250^\circ\)

      Because this angle is less than \(270^\circ\) and more than \(180^\circ\) , it terminates in Quadrant III.

      The reference angle is:

      \(\begin{align}
       \theta _R &= \theta - 180^\circ  \\
       \theta _R &= 250^\circ - 180^\circ  \\
       \theta _R &= 70^\circ  \\
       \end{align}\)

  2. Determine all the angles in standard position, for \(0^\circ \le \theta < 360^\circ \), that have a reference angle of \(15^\circ\).

    Quadrant I: \(\theta _R = \theta \)

    \(\begin{align}
     \theta _R &= \theta  \\
     15^\circ &= \theta  \\
     \end{align}\)
    Quadrant III: \(\theta _R = \theta - 180^\circ \)

    \(\begin{align}
     \theta _R &= \theta - 180^\circ  \\
     15^\circ &= \theta - 180^\circ  \\
     195^\circ &= \theta  \\
     \end{align}\)
    Quadrant II: \(\theta _R = 180^\circ - \theta \)

     \(\begin{align}
     \theta _R &= 180^\circ - \theta  \\
     15^\circ &= 180^\circ - \theta  \\
      -165^\circ &= - \theta  \\
     165^\circ &= \theta  \\
     \end{align}\)
    Quadrant IV: \(\theta _R = 360^\circ - \theta \)

    \(\begin{align}
     \theta _R &= 360^\circ - \theta  \\
     15^\circ &= 360^\circ - \theta  \\
      -345^\circ &= - \theta  \\
     345^\circ &= \theta  \\
     \end{align}\)