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 sides and angles for sine - law

In \(\triangle UVW\), let \(a = w\), \(A=31^{\circ}\), \(b = 3.3\) cm, \(B = 39^{\circ}\). According to the law of sines \(\frac{\sin(A)}{a}=\frac{\sin(B)}{b}\).

Step2: Rearrange the law - of - sines formula to solve for \(w\)

We can rewrite the formula as \(w=\frac{b\times\sin(A)}{\sin(B)}\).

Step3: Substitute the values

\(\sin(31^{\circ})\approx0.515\), \(\sin(39^{\circ})\approx0.629\), \(b = 3.3\) cm. Then \(w=\frac{3.3\times0.515}{0.629}\).

Step4: Calculate the value of \(w\)

\(3.3\times0.515 = 1.6995\), and \(\frac{1.6995}{0.629}\approx2.7\) (this is wrong. Let's correct the setup. The correct setup should be \(\frac{\sin(39^{\circ})}{w}=\frac{\sin(31^{\circ})}{3.3}\), then \(w=\frac{3.3\times\sin(39^{\circ})}{\sin(31^{\circ})}\)).
\(\sin(39^{\circ})\approx0.629\), \(\sin(31^{\circ})\approx0.515\), \(w=\frac{3.3\times0.629}{0.515}=\frac{2.0757}{0.515}\approx4.0\) cm.

Answer:

4.0 cm