Question

find all angles, ( 0^circ leq \theta < 360^circ ), that solve the following equation. (\tan\theta = -1)

Explanation:

Step1: Recall tangent values

We know that \(\tan\theta = -1\). The tangent function has a period of \(180^\circ\), and we know that \(\tan 45^\circ = 1\). For \(\tan\theta=-1\), we look at the quadrants where tangent is negative, which are the second and fourth quadrants.

Step2: Find reference angle

The reference angle \(\alpha\) for which \(\tan\alpha = 1\) is \(45^\circ\).

Step3: Find angles in second quadrant

In the second quadrant, \(\theta = 180^\circ-\alpha\). So \(\theta = 180^\circ - 45^\circ=135^\circ\).

Step4: Find angles in fourth quadrant

In the fourth quadrant, \(\theta = 360^\circ-\alpha\). So \(\theta = 360^\circ - 45^\circ = 315^\circ\). Also, we check the interval \(0^\circ\leq\theta< 360^\circ\), so \(135^\circ\) and \(315^\circ\) are the solutions.

Answer:

\(135^\circ, 315^\circ\)