#### Answer

$$RS\approx442.6yd$$

#### Work Step by Step

$$\angle R=102^\circ20'\hspace{.75cm}\angle T=32^\circ50'\hspace{.7cm}TR=582yd$$
1) Analysis
- To find $RS$, we would use law of sines. 2 groups of side and its opposite angles must be known.
- Side $TR$ is known, but its opposite angle $\angle S$ is unknown now. However, since $\angle T$ and $\angle R$ are known, $\angle S$ can be calculated using the law of the sum of 3 angles in a triangle.
- The opposite angle of $RS$, $\angle T$, is already known.
2) Calculate $\angle S$
The sum of 3 angles in a triangle is $180^\circ$.
$$\angle R+\angle T+\angle S=180^\circ$$
$$\angle S+102^\circ20'+32^\circ50'=180^\circ$$
$$\angle S+135^\circ10'=180^\circ$$
$$\angle S=44^\circ50'$$
3) Now we apply law of sines to find $RS$.
- The opposite angle of $RS$ is $\angle T$, $\sin T=\sin 32^\circ50'\approx0.54$
- $TR=582yd$, its opposite angle is $\angle S$, $\sin S=\sin 44^\circ50'\approx0.71$
According to law of sines:
$$\frac{RS}{\sin T}=\frac{TR}{\sin S}$$
$$RS=\frac{TR\sin T}{\sin S}$$
$$RS=\frac{582yd\times0.54}{0.71}$$
$$RS\approx442.6yd$$