To find [tex]\(\sin(\theta)\)[/tex] given that [tex]\(\cos(\theta) = \frac{3}{10}\)[/tex], we can use the Pythagorean identity, which states:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
1. We are given [tex]\(\cos(\theta) = \frac{3}{10}\)[/tex]. First, we calculate [tex]\(\cos^2(\theta)\)[/tex]:
[tex]\[
\cos^2(\theta) = \left(\frac{3}{10}\right)^2 = \frac{9}{100}
\][/tex]
2. Using the Pythagorean identity, we solve for [tex]\(\sin^2(\theta)\)[/tex]:
[tex]\[
\sin^2(\theta) = 1 - \cos^2(\theta) = 1 - \frac{9}{100} = \frac{100}{100} - \frac{9}{100} = \frac{91}{100}
\][/tex]
3. To find [tex]\(\sin(\theta)\)[/tex], we take the positive square root of [tex]\(\sin^2(\theta)\)[/tex] (since [tex]\(\theta\)[/tex] typically represents an angle in the first quadrant where sine is positive):
[tex]\[
\sin(\theta) = \sqrt{\frac{91}{100}} = \frac{\sqrt{91}}{10}
\][/tex]
Thus, the value of [tex]\(\sin(\theta)\)[/tex] is:
[tex]\[
\sin(\theta) = \frac{\sqrt{91}}{10}
\][/tex]
Therefore, the correct answer is:
B. [tex]\(\frac{\sqrt{91}}{10}\)[/tex]
