To determine [tex]\(\sin(\theta)\)[/tex] given that [tex]\(\cos(\theta) = \frac{3}{10}\)[/tex], we can utilize the Pythagorean identity in trigonometry, which states:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
Given:
[tex]\[
\cos(\theta) = \frac{3}{10}
\][/tex]
First, we'll find [tex]\(\cos^2(\theta)\)[/tex]:
[tex]\[
\cos^2(\theta) = \left(\frac{3}{10}\right)^2 = \frac{9}{100}
\][/tex]
Next, we substitute [tex]\(\cos^2(\theta)\)[/tex] into the Pythagorean identity to solve for [tex]\(\sin^2(\theta)\)[/tex]:
[tex]\[
\sin^2(\theta) + \frac{9}{100} = 1
\][/tex]
Subtract [tex]\(\frac{9}{100}\)[/tex] from both sides:
[tex]\[
\sin^2(\theta) = 1 - \frac{9}{100} = \frac{100}{100} - \frac{9}{100} = \frac{91}{100}
\][/tex]
Now, we take the square root of both sides to find [tex]\(\sin(\theta)\)[/tex]:
[tex]\[
\sin(\theta) = \sqrt{\frac{91}{100}} = \frac{\sqrt{91}}{10}
\][/tex]
Therefore, the correct answer is:
C. [tex]\(\frac{\sqrt{91}}{10}\)[/tex]
