Use the given information to determine the exact trigonometric value.

Given:
[tex]\[ \sin \theta = -\frac{1}{5}, \pi \ \textless \ \theta \ \textless \ \frac{3 \pi}{2} \][/tex]

Find: [tex]\(\cos \theta\)[/tex]

A. [tex]\(-\frac{2 \sqrt{6}}{5}\)[/tex]
B. [tex]\(-\frac{3 \sqrt{6}}{5}\)[/tex]
C. [tex]\(-\frac{2 \sqrt{6}}{2}\)[/tex]
D. [tex]\(-\frac{3 \sqrt{6}}{2}\)[/tex]



Answer :

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.