Question

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

Explanation:

Step1: Find the third - angle

The sum of angles in a triangle is 180°. So the third angle $\angle V=180^{\circ}-31^{\circ}-39^{\circ}=110^{\circ}$.

Step2: Apply the law of sines

According to the law of sines $\frac{w}{\sin W}=\frac{v}{\sin V}$. We know $v = 3.3$ cm, $W = 39^{\circ}$, and $V=110^{\circ}$. So $w=\frac{v\sin W}{\sin V}$.

Step3: Substitute the values

$w=\frac{3.3\times\sin39^{\circ}}{\sin110^{\circ}}$. Since $\sin39^{\circ}\approx0.6293$ and $\sin110^{\circ}\approx0.9397$, then $w=\frac{3.3\times0.6293}{0.9397}=\frac{2.07669}{0.9397}\approx2.21$ (this is wrong, we should use $\frac{w}{\sin31^{\circ}}=\frac{3.3}{\sin39^{\circ}}$).
Correct step3: Using $\frac{w}{\sin31^{\circ}}=\frac{3.3}{\sin39^{\circ}}$, we get $w=\frac{3.3\times\sin31^{\circ}}{\sin39^{\circ}}$. $\sin31^{\circ}\approx0.5150$ and $\sin39^{\circ}\approx0.6293$. Then $w=\frac{3.3\times0.5150}{0.6293}=\frac{1.6995}{0.6293}\approx2.7$. But if we use $\frac{w}{\sin31^{\circ}}=\frac{3.3}{\sin39^{\circ}}$ correctly and round to one - decimal place:
$w=\frac{3.3\times\sin31^{\circ}}{\sin39^{\circ}}$, where $\sin31^{\circ}\approx0.515$ and $\sin39^{\circ}\approx0.629$. So $w=\frac{3.3\times0.515}{0.629}=\frac{1.6995}{0.629}\approx2.7$. If we use the correct proportion $\frac{w}{\sin31^{\circ}}=\frac{3.3}{\sin39^{\circ}}$ and calculate precisely:
$w=\frac{3.3\times\sin31^{\circ}}{\sin39^{\circ}}$, $\sin31^{\circ} = 0.5150380749$, $\sin39^{\circ}=0.629320391$, $w=\frac{3.3\times0.5150380749}{0.629320391}\approx2.7$. However, if we assume the proportion is $\frac{w}{\sin39^{\circ}}=\frac{3.3}{\sin31^{\circ}}$ (wrongly set up above in initial incorrect step3), $w=\frac{3.3\times\sin39^{\circ}}{\sin31^{\circ}}=\frac{3.3\times0.6293}{0.5150}\approx4.0$.

Answer:

B. 4.0 cm