To solve for [tex]\(\sin(\theta)\)[/tex] given that [tex]\(\cos(\theta) = \frac{\sqrt{11}}{5}\)[/tex] and [tex]\(\theta\)[/tex] is an angle in quadrant I, we can use the Pythagorean identity:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
Firstly, let's substitute the given value of [tex]\(\cos(\theta)\)[/tex] into the identity:
[tex]\[
\sin^2(\theta) + \left(\frac{\sqrt{11}}{5}\right)^2 = 1
\][/tex]
Calculate [tex]\(\left(\frac{\sqrt{11}}{5}\right)^2\)[/tex]:
[tex]\[
\left(\frac{\sqrt{11}}{5}\right)^2 = \frac{11}{25}
\][/tex]
Now substitute this back into the equation:
[tex]\[
\sin^2(\theta) + \frac{11}{25} = 1
\][/tex]
To isolate [tex]\(\sin^2(\theta)\)[/tex], subtract [tex]\(\frac{11}{25}\)[/tex] from both sides:
[tex]\[
\sin^2(\theta) = 1 - \frac{11}{25}
\][/tex]
Convert 1 into a fraction with the same denominator:
[tex]\[
1 = \frac{25}{25}
\][/tex]
Subtract the fractions:
[tex]\[
\sin^2(\theta) = \frac{25}{25} - \frac{11}{25} = \frac{14}{25}
\][/tex]
Take the square root of both sides to solve for [tex]\(\sin(\theta)\)[/tex]:
[tex]\[
\sin(\theta) = \pm \sqrt{\frac{14}{25}} = \pm \frac{\sqrt{14}}{5}
\][/tex]
Since [tex]\(\theta\)[/tex] is in quadrant I, where sine is positive:
[tex]\[
\sin(\theta) = \frac{\sqrt{14}}{5}
\][/tex]
Thus, the correct answer is:
A. [tex]\(\frac{\sqrt{14}}{5}\)[/tex]
