Given that $\cos \left(\theta \right)$ = $\frac{3}{10}$, what is the value of $\sin \left(\theta \right)$ if $\theta$ is an acute angle?

• A. $\frac{3}{\sqrt{10}}$
• B. $\frac{\sqrt{91}}{10}$
• C. $\frac{7}{10}$
• D. $\frac{\sqrt{19}}{10}$

Answer: (B)

Verified step by step guidance

1. Identify the given information: \( \cos(\theta) = \frac{3}{10} \) and \( \theta \) is an acute angle (meaning \( 0 < \theta < 90^\circ \)).
2. Recall the Pythagorean identity relating sine and cosine: \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
3. Substitute the given cosine value into the identity: \( \sin^2(\theta) + \left(\frac{3}{10}\right)^2 = 1 \).
4. Simplify the equation to solve for \( \sin^2(\theta) \): \( \sin^2(\theta) = 1 - \frac{9}{100} \).
5. Since \( \theta \) is acute, \( \sin(\theta) \) is positive, so take the positive square root: \( \sin(\theta) = \sqrt{1 - \frac{9}{100}} \). Simplify the expression under the square root to find \( \sin(\theta) \).