Question

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

Explanation:

Step1: Match angles to sides

In $\triangle UVW$, angle $\angle U = 31^\circ$, opposite side $VW = 3.3$ cm; angle $\angle W = 39^\circ$, opposite side $w = UV$. By the law of sines:
$$\frac{w}{\sin(\angle W)} = \frac{VW}{\sin(\angle U)}$$

Step2: Rearrange to solve for $w$

Isolate $w$ by multiplying both sides by $\sin(\angle W)$:
$$w = \frac{VW \cdot \sin(\angle W)}{\sin(\angle U)}$$

Step3: Substitute values and calculate

Substitute $VW=3.3$, $\angle W=39^\circ$, $\angle U=31^\circ$:
$$w = \frac{3.3 \cdot \sin(39^\circ)}{\sin(31^\circ)}$$
Using $\sin(39^\circ)\approx0.6293$, $\sin(31^\circ)\approx0.5150$:
$$w \approx \frac{3.3 \cdot 0.6293}{0.5150} \approx \frac{2.0767}{0.5150} \approx 4.0$$

Answer:

4.0 cm