To find the cosine of [tex]\(\theta\)[/tex] given that [tex]\(\sin \theta = -\frac{3 \sqrt{13}}{13}\)[/tex] and [tex]\(\theta\)[/tex] is in the fourth quadrant, we need to follow these steps:
1. Understand the Positional Context:
[tex]\(\theta\)[/tex] is in the fourth quadrant. In this quadrant, the sine function is negative, and the cosine function is positive.
2. Calculate [tex]\(\sin^2 \theta\)[/tex]:
We are given [tex]\(\sin \theta = -\frac{3 \sqrt{13}}{13}\)[/tex]. First, square this value:
[tex]\[
\sin^2 \theta = \left(-\frac{3 \sqrt{13}}{13}\right)^2 = \left(\frac{3 \sqrt{13}}{13}\right)^2 = \frac{9 \cdot 13}{169} = \frac{117}{169}
\][/tex]
3. Use the Pythagorean Identity:
Recall the Pythagorean identity [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex]. Using the value found for [tex]\(\sin^2 \theta\)[/tex]:
[tex]\[
\cos^2 \theta = 1 - \sin^2 \theta = 1 - \frac{117}{169} = \frac{169 - 117}{169} = \frac{52}{169} = \frac{4 \cdot 13}{13^2} = \frac{4}{13}
\][/tex]
4. Find [tex]\(\cos \theta\)[/tex]:
To find [tex]\(\cos \theta\)[/tex], we take the square root of [tex]\(\cos^2 \theta\)[/tex]. Since [tex]\(\theta\)[/tex] is in the fourth quadrant where cosine is positive, we select the positive root:
[tex]\[
\cos \theta = \sqrt{\frac{4}{13}} = \frac{2}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}} = \frac{2 \sqrt{13}}{13}
\][/tex]
5. Choose the Correct Answer:
Based on the options provided:
[tex]\[
\text{A. } \frac{2 \sqrt{13}}{13}
\][/tex]
B. [tex]\(-\frac{3}{2}\)[/tex]
C. [tex]\(-\frac{2 \sqrt{13}}{13}\)[/tex]
D. [tex]\(-\frac{2}{3}\)[/tex]
The correct answer is:
[tex]\[
\boxed{\frac{2 \sqrt{13}}{13}}
\][/tex]
