# Chapter 6 - Section 6.1 - The Law of Sines - Exercise Set - Page 720: 1

The required values are $B=42{}^\circ,a\approx 8.1,$ and $b\approx 8.1$.

#### Work Step by Step

At first we will find B. Properties of a triangle: Sum of three angles is $A+B+C=180{}^\circ$ \begin{align} & A+B+C=180{}^\circ \\ & 42{}^\circ +B+96{}^\circ =180{}^\circ \\ & B=180{}^\circ -138{}^\circ \\ & B=42{}^\circ \end{align} Now, to find the remaining sides, we will use the ratio $\frac{c}{\sin C}$ or $\frac{12}{\sin 96{}^\circ }$. Now, we will use the law of sines to find a. \begin{align} & \frac{a}{\sin A}=\frac{c}{\sin C} \\ & \frac{a}{\sin 42{}^\circ }=\frac{12}{\sin 96{}^\circ } \\ & a=\frac{12\sin 42{}^\circ }{\sin 96{}^\circ } \\ & a=8.1 \end{align} Again, use the law of sines to find b \begin{align} & \frac{b}{\sin B}=\frac{c}{\sin C} \\ & \frac{b}{\sin 42{}^\circ }=\frac{12}{\sin 96{}^\circ } \\ & b=\frac{12\sin 42{}^\circ }{\sin 96{}^\circ } \\ & b=8.1 \end{align} The solution is $B=42{}^\circ,a\approx 8.1,$ and $b\approx 8.1$.

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