In a right triangle, angle $J$ has opposite side length $20$ and adjacent side length $21$. What is the value of $\sin J$?

• A. $\frac{29}{20}$
• B. $\frac{20}{21}$
• C. $\frac{21}{29}$
• D. $\frac{20}{29}$

Answer: (D)

Verified step by step guidance

1. Identify the sides of the right triangle relative to angle J: the opposite side length is 20, and the adjacent side length is 21.
2. Recall the definition of sine for an angle in a right triangle: \(\sin J = \frac{\text{opposite side}}{\text{hypotenuse}}\).
3. Use the Pythagorean theorem to find the hypotenuse length: \(\text{hypotenuse} = \sqrt{(\text{opposite})^2 + (\text{adjacent})^2} = \sqrt{20^2 + 21^2}\).
4. Calculate the hypotenuse expression without simplifying the square root: \(\sqrt{400 + 441} = \sqrt{841}\).
5. Write the sine of angle J as \(\sin J = \frac{20}{\sqrt{841}}\), which simplifies to \(\sin J = \frac{20}{29}\).