To find the exact value of [tex]\(\cos \theta\)[/tex] given the information [tex]\(\sin \theta = -\frac{1}{5}\)[/tex] and [tex]\(\pi < \theta < \frac{3\pi}{2}\)[/tex]:
1. Identify the quadrant: Based on the given range [tex]\(\pi < \theta < \frac{3\pi}{2}\)[/tex], we know [tex]\(\theta\)[/tex] is in the third quadrant. In this quadrant, both sine and cosine values are negative.
2. Use the Pythagorean identity: Recall the identity [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex]. Substitute the given sine value into the equation:
[tex]\[
\left(-\frac{1}{5}\right)^2 + \cos^2 \theta = 1
\][/tex]
Simplify:
[tex]\[
\frac{1}{25} + \cos^2 \theta = 1
\][/tex]
[tex]\[
\cos^2 \theta = 1 - \frac{1}{25}
\][/tex]
[tex]\[
\cos^2 \theta = \frac{25}{25} - \frac{1}{25}
\][/tex]
[tex]\[
\cos^2 \theta = \frac{24}{25}
\][/tex]
3. Solve for [tex]\(\cos \theta\)[/tex]: Since [tex]\(\cos \theta\)[/tex] can be positive or negative, but we know from the quadrant (third quadrant) that [tex]\(\cos \theta\)[/tex] is negative:
[tex]\[
\cos \theta = -\sqrt{\frac{24}{25}}
\][/tex]
Simplify the square root:
[tex]\[
\cos \theta = -\frac{\sqrt{24}}{5}
\][/tex]
Further simplify [tex]\(\sqrt{24}\)[/tex]:
[tex]\[
\sqrt{24} = \sqrt{4 \cdot 6} = \sqrt{4} \cdot \sqrt{6} = 2\sqrt{6}
\][/tex]
Therefore:
[tex]\[
\cos \theta = -\frac{2\sqrt{6}}{5}
\][/tex]
Hence, the exact value of [tex]\(\cos \theta\)[/tex] is [tex]\(-\frac{2\sqrt{6}}{5}\)[/tex].
4. Match with the choices:
The correct answer is:
- (a) [tex]\(-\frac{2 \sqrt{6}}{5}\)[/tex]
This matches our calculated value.
