To find the values of \(\theta\) where \(0^{\circ} \leq \theta \leq 360^{\circ}\) and \(\cos \theta = -\frac{\sqrt{3}}{2}\), we can follow these steps:
1. Identify the cosine value:
[tex]\[
\cos \theta = -\frac{\sqrt{3}}{2}
\][/tex]
2. Understand the key angles where cosine takes on specific values:
[tex]\[
\cos(\theta) = -\frac{\sqrt{3}}{2}
\][/tex]
The reference angle for \(\frac{\sqrt{3}}{2}\) is \(30^{\circ}\) because \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\).
3. Since the cosine function is negative, consider in which quadrants the cosine function is negative:
- Cosine is negative in the second and third quadrants.
4. Calculate the angles in these quadrants given that the reference angle is \(30^{\circ}\):
- In the second quadrant, the angle is found by subtracting the reference angle from \(180^{\circ}\):
[tex]\[
180^{\circ} - 30^{\circ} = 150^{\circ}
\][/tex]
- In the third quadrant, the angle is found by adding the reference angle to \(180^{\circ}\):
[tex]\[
180^{\circ} + 30^{\circ} = 210^{\circ}
\][/tex]
Therefore, the angles where \(\cos \theta = -\frac{\sqrt{3}}{2}\) and \(0^{\circ} \leq \theta \leq 360^{\circ}\) are:
[tex]\[
\theta = 150^{\circ} \quad \text{and} \quad \theta = 210^{\circ}
\][/tex]
We can write our final answers in increasing order as:
[tex]\[
\begin{array}{ll}
\theta = 150^{\circ} \\
\theta = 210^{\circ}
\end{array}
\][/tex]
