To determine \(\cos(\theta)\) given that \(\sin(\theta) = -\frac{1}{3}\) and \(\pi < \theta < \frac{3\pi}{2}\), follow these steps:
1. Identify the Quadrant:
Since \(\pi < \theta < \frac{3\pi}{2}\), the angle \(\theta\) is in the third quadrant. In the third quadrant, sine is negative and cosine is also negative.
2. Use the Pythagorean Identity:
The Pythagorean identity states:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
3. Substitute the Known Sine Value:
Given \(\sin(\theta) = -\frac{1}{3}\), first find \(\sin^2(\theta)\):
[tex]\[
\sin^2(\theta) = \left(-\frac{1}{3}\right)^2 = \frac{1}{9}
\][/tex]
4. Express \(\cos^2(\theta)\) in Terms of Known Values:
We use the identity \(\sin^2(\theta) + \cos^2(\theta) = 1\) to solve for \(\cos^2(\theta)\):
[tex]\[
\cos^2(\theta) = 1 - \sin^2(\theta)
\][/tex]
Substituting \(\sin^2(\theta)\):
[tex]\[
\cos^2(\theta) = 1 - \frac{1}{9}
\][/tex]
Simplify the expression:
[tex]\[
\cos^2(\theta) = \frac{9}{9} - \frac{1}{9} = \frac{8}{9}
\][/tex]
5. Determine \(\cos(\theta)\):
To find \(\cos(\theta)\), take the square root of both sides. Since \(\cos(\theta)\) is in the third quadrant and must be negative:
[tex]\[
\cos(\theta) = -\sqrt{\frac{8}{9}} = -\frac{\sqrt{8}}{3} = -\frac{2\sqrt{2}}{3}
\][/tex]
Therefore, the value of \(\cos(\theta)\) is:
[tex]\[
\boxed{-\frac{2\sqrt{2}}{3}}
\][/tex]
