In right triangle QRS, if angle $Q$ is the right angle and side $QR$ has length $6$ and side $QS$ has length $10$, what is the value of $\sin \left(S\right)$?

• A. $\frac{6}{10}$
• B. $\frac{10}{6}$
• C. $\frac{8}{10}$
• D. $\frac{6}{8}$

Answer: (A)

Verified step by step guidance

1. Identify the right angle in triangle QRS, which is angle Q. This means side QR and side QS are the legs meeting at the right angle, and side RS is the hypotenuse.
2. Recall that the sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse: \(\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}\).
3. Determine which side is opposite angle S. Since angle Q is the right angle, side QR is opposite angle S, and side QS is adjacent to angle S.
4. Calculate the length of the hypotenuse RS using the Pythagorean theorem: \(RS = \sqrt{QR^2 + QS^2} = \sqrt{6^2 + 10^2}\).
5. Express \(\sin(S)\) as the ratio of the length of the side opposite angle S (which is QR) over the hypotenuse RS: \(\sin(S) = \frac{QR}{RS} = \frac{6}{\sqrt{6^2 + 10^2}}\).