Example  1
Solve for \(\theta\).

  1. \(\cos \theta = \frac{1}{2}, \thinspace 0^\circ \le \theta < 180^\circ \)

    Step 1: Determine in which quadrant(s) the solution(s) will lie.

    Recall that cos \(\theta\) is positive in Quadrants I and IV. Because \(\theta\) is restricted to Quadrants I and II, the only answer you will be concerned with is the one in Quadrant I.

    Step 2: Determine the reference angle that corresponds to the trigonometric ratio.

    Notice that cos \(\theta = \frac{1}{2}\) is a special triangle ratio, specifically when \(\theta = 60^\circ\). Therefore, the reference angle is \(60^\circ\).

    Step 3: Sketch the reference angle in the appropriate quadrant, and then determine the angle in standard position.

    In this case, the reference angle is equal to the angle in standard position because the angle terminates in Quadrant I.

    Therefore, \(\theta = 60^\circ\).


  2. \(\tan \theta = -1, \thinspace 0^\circ \le \theta < 360^\circ \)

    Step 1: Determine in which quadrant(s) the solution(s) will lie.

    Recall that tan \(\theta\) is negative in Quadrants II and IV.

    Step 2: Determine the reference angle that corresponds to the positive trigonometric ratio.

    Notice that tan \(\theta = 1\) is a special triangle ratio, specifically when \(\theta = 45^\circ\). Therefore, the reference angle is \(45^\circ\).

    Step 3: Sketch the reference angles in the appropriate quadrants, and then determine the angles in standard position.

    Quadrant II:

    \(\begin{align}
     180^\circ - \theta &= \theta _R  \\
     180^\circ - \theta &= 45^\circ  \\
     135^\circ &= \theta  \\
     \end{align}\)

    Quadrant IV:

    \(\begin{align}
     360^\circ - \theta &= \theta _R  \\
     360^\circ - \theta &= 45^\circ  \\
     315^\circ &= \theta  \\
     \end{align}\)

    Therefore, \(\theta = 135^\circ \thinspace \rm{and}\thinspace \theta = 315^\circ \)