Question

find all angles, ( 0^circ leq \theta < 360^circ ), that solve the following equation. (\tan \theta = -\frac{sqrt{3}}{3}) answer (\theta = ) additional solution no solution submit answer show examples

Explanation:

Step1: Recall tangent values

We know that \(\tan 30^\circ=\frac{\sqrt{3}}{3}\), so \(\tan\theta = -\frac{\sqrt{3}}{3}\) means \(\theta\) is in quadrants where tangent is negative, i.e., QII and QIV.

Step2: Find reference angle

The reference angle \(\alpha\) where \(\tan\alpha=\frac{\sqrt{3}}{3}\) is \(30^\circ\).

Step3: Calculate angles in QII and QIV

• For QII: \(\theta = 180^\circ - 30^\circ = 150^\circ\)
• For QIV: \(\theta = 360^\circ - 30^\circ = 330^\circ\)

Answer:

\(150^\circ, 330^\circ\)