We are given:
[tex]\[
\sin (\theta) = -\frac{4 \sqrt{29}}{29}
\][/tex]
and we need to find the value of [tex]\(\cos (\theta)\)[/tex] given that [tex]\(\theta\)[/tex] is in quadrant IV.
First, let's recall that in quadrant IV, the sine function is negative, and the cosine function is positive. This is consistent with our given value of [tex]\(\sin(\theta)\)[/tex].
Next, we use the Pythagorean identity, which states:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
Substitute the given [tex]\(\sin (\theta)\)[/tex]:
[tex]\[
\sin(\theta) = -\frac{4 \sqrt{29}}{29}
\][/tex]
Calculate [tex]\(\sin^2 (\theta)\)[/tex]:
[tex]\[
\sin^2(\theta) = \left(-\frac{4 \sqrt{29}}{29}\right)^2 = \frac{(4 \sqrt{29})^2}{29^2} = \frac{16 \cdot 29}{841} = \frac{464}{841}
\][/tex]
Next, we use the Pythagorean identity to find [tex]\(\cos^2 (\theta)\)[/tex]:
[tex]\[
\cos^2(\theta) = 1 - \sin^2(\theta) = 1 - \frac{464}{841} = \frac{841}{841} - \frac{464}{841} = \frac{377}{841}
\][/tex]
Now, take the square root of both sides to find [tex]\(\cos (\theta)\)[/tex]. Since [tex]\(\theta\)[/tex] is in quadrant IV, [tex]\(\cos (\theta)\)[/tex] is positive:
[tex]\[
\cos(\theta) = \sqrt{\frac{377}{841}} = \frac{\sqrt{377}}{29}
\][/tex]
The correct answer is:
B. [tex]\(\frac{\sqrt{377}}{29}\)[/tex]
