Question

find the sine of $\angle v$. simplify your answer and write it as a proper fraction, improper fraction, or whole number. $\sin(v) = $

Explanation:

Step1: Identify triangle type and sides

This is a right triangle with right angle at \( X \). Hypotenuse \( VW = 89 \), leg \( VX = 39 \). First, find the other leg \( XW \) using Pythagoras: \( XW=\sqrt{VW^{2}-VX^{2}}=\sqrt{89^{2}-39^{2}} \).
\[

$$\begin{align*} 89^{2}-39^{2}&=(89 - 39)(89 + 39)\\ &=50\times128\\ &=6400 \end{align*}$$

\]
So \( XW=\sqrt{6400}=80 \).

Step2: Recall sine definition

In right triangle, \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \). For \( \angle V \), opposite side is \( XW = 80 \), hypotenuse \( VW = 89 \). Thus, \( \sin(V)=\frac{XW}{VW}=\frac{80}{89} \).

Answer:

\(\frac{80}{89}\)