Question

use the law of sines to find the value of w. what is the best approximation of the value of w? law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$

Explanation:

Step1: Identify the angles and side for law - of - sines

In $\triangle UVW$, we know $\angle U = 31^{\circ}$, $\angle W=39^{\circ}$, and side $v = 3.3$ cm. We want to find side $w$. First, find $\angle V=180^{\circ}-(31^{\circ}+39^{\circ}) = 110^{\circ}$.

Step2: Apply the law of sines

The law of sines is $\frac{\sin U}{u}=\frac{\sin V}{v}=\frac{\sin W}{w}$. We use $\frac{\sin U}{u}=\frac{\sin W}{w}$, so $w=\frac{v\sin W}{\sin U}$.

Step3: Substitute the values

Substitute $v = 3.3$ cm, $\angle U = 31^{\circ}$, and $\angle W = 39^{\circ}$ into the formula. $\sin31^{\circ}\approx0.515$, $\sin39^{\circ}\approx0.629$. Then $w=\frac{3.3\times0.629}{0.515}=\frac{2.0757}{0.515}\approx4.0$ cm.

Answer:

$4.0$ cm