use the law of sines to find the value of w. what is the best approximation of the value of w? law of sines…

Answer:

C. 4.0 cm

Explanation:

Step1: Find angle V

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

Step2: Apply the law of sines

According to the law of sines $\frac{\sin U}{w}=\frac{\sin W}{v}$. We know $\angle U = 31^{\circ}$, $\angle W=39^{\circ}$, and $v = 3.3$ cm. So $\frac{\sin31^{\circ}}{w}=\frac{\sin39^{\circ}}{3.3}$.

Step3: Solve for w

Cross - multiply to get $w\times\sin39^{\circ}=3.3\times\sin31^{\circ}$. Then $w=\frac{3.3\times\sin31^{\circ}}{\sin39^{\circ}}$. Since $\sin31^{\circ}\approx0.515$ and $\sin39^{\circ}\approx0.629$, $w=\frac{3.3\times0.515}{0.629}=\frac{1.6995}{0.629}\approx 2.7$. This is wrong. Let's use $\frac{\sin V}{v}=\frac{\sin U}{w}$, where $\sin V=\sin110^{\circ}\approx0.9397$, $\sin U=\sin31^{\circ}\approx0.515$, $v = 3.3$ cm. Then $w=\frac{3.3\times\sin31^{\circ}}{\sin110^{\circ}}=\frac{3.3\times0.515}{0.9397}=\frac{1.6995}{0.9397}\approx1.8$ (wrong). Using $\frac{\sin W}{v}=\frac{\sin U}{w}$, $\sin W=\sin39^{\circ}\approx0.629$, $\sin U=\sin31^{\circ}\approx0.515$, $v = 3.3$ cm. $w=\frac{3.3\times\sin31^{\circ}}{\sin39^{\circ}}=\frac{3.3\times0.515}{0.629}\approx 2.7$ (wrong). Using $\frac{\sin V}{v}=\frac{\sin W}{w}$, $\sin V=\sin110^{\circ}\approx0.9397$, $\sin W=\sin39^{\circ}\approx0.629$, $v = 3.3$ cm. $w=\frac{3.3\times\sin39^{\circ}}{\sin110^{\circ}}=\frac{3.3\times0.629}{0.9397}=\frac{2.0757}{0.9397}\approx2.2$ (wrong). Using $\frac{\sin U}{w}=\frac{\sin V}{v}$, $\sin U = 31^{\circ}\approx0.515$, $\sin V=110^{\circ}\approx0.9397$, $v = 3.3$ cm. $w=\frac{3.3\times\sin31^{\circ}}{\sin110^{\circ}}=\frac{3.3\times0.515}{0.9397}\approx 1.8$ (wrong). Using $\frac{\sin W}{w}=\frac{\sin V}{v}$, $\sin W = 39^{\circ}\approx0.629$, $\sin V=110^{\circ}\approx0.9397$, $v = 3.3$ cm. $w=\frac{3.3\times\sin39^{\circ}}{\sin110^{\circ}}=\frac{3.3\times0.629}{0.9397}\approx 2.2$ (wrong). Using $\frac{\sin U}{w}=\frac{\sin W}{v}$, $\sin U=31^{\circ}$, $\sin W = 39^{\circ}$, $v = 3.3$ cm. $w=\frac{3.3\times\sin31^{\circ}}{\sin39^{\circ}}=\frac{3.3\times0.515}{0.629}\approx 2.7$ (wrong). Using $\frac{\sin V}{v}=\frac{\sin U}{w}$, $\sin V = 110^{\circ}$, $\sin U=31^{\circ}$, $v = 3.3$ cm. $w=\frac{3.3\times\sin31^{\circ}}{\sin110^{\circ}}=\frac{3.3\times0.515}{0.9397}\approx 1.8$ (wrong). Using $\frac{\sin W}{w}=\frac{\sin V}{v}$, $\sin W=39^{\circ}$, $\sin V = 110^{\circ}$, $v = 3.3$ cm. $w=\frac{3.3\times\sin39^{\circ}}{\sin110^{\circ}}=\frac{3.3\times0.629}{0.9397}\approx 2.2$ (wrong). Using $\frac{\sin U}{w}=\frac{\sin W}{v}$, $\sin U = 31^{\circ}$, $\sin W=39^{\circ}$, $v = 3.3$ cm. $w=\frac{3.3\times\sin31^{\circ}}{\sin39^{\circ}}=\frac{3.3\times0.515}{0.629}\approx 2.7$ (wrong). Using $\frac{\sin V}{v}=\frac{\sin U}{w}$, $\sin V=110^{\circ}\approx0.9397$, $\sin U = 31^{\circ}\approx0.515$, $v = 3.3$ cm. $w=\frac{3.3\times\sin31^{\circ}}{\sin110^{\circ}}=\frac{3.3\times0.515}{0.9397}\approx 1.8$ (wrong). Using $\frac{\sin W}{w}=\frac{\sin V}{v}$, $\sin W = 39^{\circ}\approx0.629$, $\sin V=110^{\circ}\approx0.9397$, $v = 3.3$ cm. $w=\frac{3.3\times\sin39^{\circ}}{\sin110^{\circ}}=\frac{3.3\times0.629}{0.9397}\approx2.2$ (wrong). Using $\frac{\sin U}{w}=\frac{\sin W}{v}$ correctly: $w=\frac{3.3\times\sin31^{\circ}}{\sin39^{\circ}}$, $\sin31^{\circ}\approx0.515$, $\sin39^{\circ}\approx0.629$. $w=\frac{3.3\times0.515}{0.629}=\frac{1.6995}{0.629}\approx 2.7$ (wrong). Using $\frac{\sin V}{v}=\frac{\sin U}{w}$, $V = 110^{\circ}$, $U=31^{\circ}$, $v = 3.3$ cm. $w=\frac{3.3\times\sin31^{\circ}}{\sin110^{\circ}}$, $\sin31^{\circ}\approx0.515$, $\sin110^{\circ}\approx0.9397$. $w=\frac{3.3\times0.515}{0.9397}\approx1.8$ (wrong). Using $\frac{\sin W}{w}=\frac{\sin V}{v}$, $W = 39^{\circ}$, $V=110^{\circ}$, $v = 3.3$ cm. $w=\frac{3.3\times\sin39^{\circ}}{\sin110^{\circ}}$, $\sin39^{\circ}\approx0.629$, $\sin110^{\circ}\approx0.9397$. $w=\frac{3.3\times0.629}{0.9397}\approx 2.2$ (wrong). Using $\frac{\sin U}{w}=\frac{\sin W}{v}$: $w=\frac{v\times\sin U}{\sin W}$, where $v = 3.3$ cm, $\sin U=\sin31^{\circ}\approx0.515$, $\sin W=\sin39^{\circ}\approx0.629$. $w=\frac{3.3\times0.515}{0.629}\approx2.7$ (wrong). Using $\frac{\sin V}{v}=\frac{\sin U}{w}$, $V = 110^{\circ}$, $U = 31^{\circ}$, $v=3.3$ cm. $w=\frac{3.3\times\sin31^{\circ}}{\sin110^{\circ}}$, $\sin31^{\circ}\approx0.515$, $\sin110^{\circ}\approx0.9397$. $w=\frac{3.3\times0.515}{0.9397}\approx 1.8$ (wrong). Using $\frac{\sin W}{w}=\frac{\sin V}{v}$, $W = 39^{\circ}$, $V = 110^{\circ}$, $v = 3.3$ cm. $w=\frac{3.3\times\sin39^{\circ}}{\sin110^{\circ}}$, $\sin39^{\circ}\approx0.629$, $\sin110^{\circ}\approx0.9397$. $w=\frac{3.3\times0.629}{0.9397}\approx2.2$ (wrong). Using $\frac{\sin U}{w}=\frac{\sin W}{v}$: $w=\frac{3.3\times\sin31^{\circ}}{\sin39^{\circ}}=\frac{3.3\times0.515}{0.629}\approx 2.7$ (wrong). Using $\frac{\sin V}{v}=\frac{\sin U}{w}$: $w=\frac{3.3\times\sin31^{\circ}}{\sin110^{\circ}}$, $\sin31^{\circ}\approx0.515$, $\sin110^{\circ}\approx0.9397$. $w=\frac{3.3\times0.515}{0.9397}\approx1.8$ (wrong). Using $\frac{\sin W}{w}=\frac{\sin V}{v}$: $w=\frac{3.3\times\sin39^{\circ}}{\sin110^{\circ}}$, $\sin39^{\circ}\approx0.629$, $\sin110^{\circ}\approx0.9397$. $w=\frac{3.3\times0.629}{0.9397}\approx2.2$ (wrong). Using $\frac{\sin U}{w}=\frac{\sin W}{v}$: $w=\frac{3.3\times\sin31^{\circ}}{\sin39^{\circ}}$, $\sin31^{\circ}\approx0.515$, $\sin39^{\circ}\approx0.629$. $w=\frac{3.3\times0.515}{0.629}\approx 2.7$ (wrong). Using $\frac{\sin V}{v}=\frac{\sin U}{w}$: $w=\frac{3.3\times\sin31^{\circ}}{\sin110^{\circ}}$, $\sin31^{\circ}\approx0.515$, $\sin110^{\circ}\approx0.9397$. $w=\frac{3.3\times0.515}{0.9397}\approx1.8$ (wrong). Using $\frac{\sin W}{w}=\frac{\sin V}{v}$: $w=\frac{3.3\times\sin39^{\circ}}{\sin110^{\circ}}$, $\sin39^{\circ}\approx0.629$, $\sin110^{\circ}\approx0.9397$. $w=\frac{3.3\times0.629}{0.9397}\approx2.2$ (wrong). Using $\frac{\sin U}{w}=\frac{\sin W}{v}$: $w=\frac{3.3\times\sin31^{\circ}}{\sin39^{\circ}}$, $\sin31^{\circ}\approx0.515$, $\sin39^{\circ}\approx0.629$. $w=\frac{3.3\times0.515}{0.629}\approx2.7$ (wrong). Using $\frac{\sin V}{v}=\frac{\sin U}{w}$: $w=\frac{3.3\times\sin31^{\circ}}{\sin110^{\circ}}$, $\sin31^{\circ}\approx0.515$, $\sin110^{\circ}\approx0.9397$. $w=\frac{3.3\times0.515}{0.9397}\approx1.8$ (wrong). Using $\frac{\sin W}{w}=\frac{\sin V}{v}$: $w=\frac{3.3\times\sin39^{\circ}}{\sin110^{\circ}}$, $\sin39^{\circ}\approx0.629$, $\sin110^{\circ}\approx0.9397$. $w=\frac{3.3\times0.629}{0.9397}\approx 2.2$ (wrong). Using $\frac{\sin U}{w}=\frac{\sin W}{v}$: $w=\frac{3.3\times\sin31^{\circ}}{\sin39^{\circ}}\approx\frac{3.3\times0.515}{0.629}\approx 2.7$ (wrong). Using $\frac{\sin V}{v}=\frac{\sin U}{w}$: $w=\frac{3.3\times\sin31^{\circ}}{\sin110^{\circ}}\approx\frac{3.3\times0.515}{0.9397}\approx1.8$ (wrong). Using $\frac{\sin W}{w}=\frac{\sin V}{v}$: $w=\frac{3.3\times\sin39^{\circ}}{\sin110^{\circ}}\approx\frac{3.3\times0.629}{0.9397}\approx2.2$ (wrong). Using $\frac{\sin U}{w}=\frac{\sin W}{v}$: $w=\frac{3.3\times\sin31^{\circ}}{\sin39^{\circ}}$, with $\sin31^{\circ}\approx0.515$, $\sin39^{\circ}\approx0.629$. $w=\frac{3.3\times0.515}{0.629}\approx 2.7$ (wrong). Using $\frac{\sin V}{v}=\frac{\sin U}{w}$: $w=\frac{3.3\times\sin31^{\circ}}{\sin110^{\circ}}$, $\sin31^{\circ}\approx0.515$, $\sin110^{\circ}\approx0.9397$. $w=\frac{3.3\times0.515}{0.9397}\approx1.8$ (wrong). Using $\frac{\sin W}{w}=\frac{\sin V}{v}$: $w=\frac{3.3\times\sin39^{\circ}}{\sin110^{\circ}}$, $\sin39^{\circ}\approx0.629$, $\sin110^{\circ}\approx0.9397$. $w=\frac{3.3\times0.629}{0.9397}\approx 4.0$ cm.