Example  1
Determine the reference angle for each angle in standard position.

  1. \(145^\circ\)

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

    The reference angle is:

    \(\begin{align}
     \theta _R &= 180^\circ - \theta  \\
     \theta _R &= 180^\circ - 145^\circ  \\
     \theta _R &= 35^\circ  \\
     \end{align}\)

  2. \(320^\circ\)

    Because this angle is less than \(360^\circ\) and more than \(270^\circ\), it terminates in Quadrant IV.

    The reference angle is:

    \(\begin{align}
     \theta _R &= 360^\circ - \theta  \\
     \theta _R &= 360^\circ - 320^\circ  \\
     \theta _R &= 40^\circ  \\
     \end{align}\)

  3. \(28^\circ\)

    Because this angle is less than \(90^\circ\) and more than \(0^\circ\), it terminates in Quadrant I.

    The reference angle is:

    \(\begin{align}
     \theta _R &= \theta  \\
     \theta _R &= 28^\circ  \\
     \end{align}\)

  4. \(190^\circ\)

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

    The reference angle is:

    \(\begin{align}
     \theta _R &= \theta - 180^\circ  \\
     \theta _R &= 190^\circ - 180^\circ  \\
     \theta _R &= 10^\circ  \\
     \end{align}\)