To determine [tex]\(\sin (\theta)\)[/tex] given that [tex]\(\cos (\theta) = \frac{3}{10}\)[/tex], we will use the Pythagorean identity:
[tex]\[
\sin^2 (\theta) + \cos^2 (\theta) = 1
\][/tex]
Given:
[tex]\[
\cos (\theta) = \frac{3}{10}
\][/tex]
First, we square the cosine value:
[tex]\[
\cos^2 (\theta) = \left(\frac{3}{10}\right)^2 = \frac{9}{100}
\][/tex]
Next, we use the Pythagorean identity to find [tex]\(\sin^2 (\theta)\)[/tex]:
[tex]\[
\sin^2 (\theta) = 1 - \cos^2 (\theta)
\][/tex]
[tex]\[
\sin^2 (\theta) = 1 - \frac{9}{100} = \frac{100}{100} - \frac{9}{100} = \frac{91}{100}
\][/tex]
Now, to find [tex]\(\sin (\theta)\)[/tex], we take the square root of both sides:
[tex]\[
\sin (\theta) = \sqrt{\sin^2 (\theta)}
\][/tex]
[tex]\[
\sin (\theta) = \sqrt{\frac{91}{100}} = \frac{\sqrt{91}}{10}
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{\frac{\sqrt{91}}{10}}
\][/tex]
So, the correct answer is option D:
D. [tex]\(\frac{\sqrt{91}}{10}\)[/tex]
