Answer
There is one triangle and the solution is $C=101{}^\circ,B=37{}^\circ,c=92.4$
Work Step by Step
The provided angles and sides of the triangle are provided below:
$A=42{}^\circ,a=63,b=57$
Using the law of sine we will find the angle B of the triangle. That is,
$\begin{align}
& \frac{\sin A}{a}=\frac{\sin B}{b} \\
& \frac{\sin 42{}^\circ }{63}=\frac{\operatorname{sinB}}{57} \\
& \operatorname{sinB}=57\times \frac{\sin 42{}^\circ }{63} \\
& =0.6054
\end{align}$
For this there are two angles possible, that is,
$\begin{align}
& {{B}_{1}}=37{}^\circ \\
& {{B}_{2}}=180{}^\circ -37{}^\circ \\
& =143{}^\circ
\end{align}$
But ${{B}_{2}}$ is nota possible angle because $42{}^\circ +143{}^\circ =185{}^\circ $. Therefore, the possible angle is ${{B}_{1}}$
The angle C will be determined as below:
$\begin{align}
& C=180-{{B}_{1}}-A \\
& =180{}^\circ -37{}^\circ -42{}^\circ \\
& =101{}^\circ
\end{align}$
Using the law of sines we will find c, that is:
$\begin{align}
& \frac{c}{\sin C}=\frac{a}{\operatorname{sinA}} \\
& \frac{c}{\sin 101{}^\circ }=\frac{63}{\sin 42{}^\circ } \\
& c=\sin 101{}^\circ \times \frac{63}{\sin 42{}^\circ } \\
& =92.4
\end{align}$
As the values of the measurements of the triangle are unique, therefore, there is one triangle and the measurement of the triangle are: ${{B}_{1}}\left( B \right)=37{}^\circ,C=101{}^\circ,\ \text{ and }\ c=92.4$
