## Precalculus (6th Edition) Blitzer

The two possible set of values are $A=59{}^\circ,C=84{}^\circ \text{ and }c\approx 14.4\text{ or }A=121{}^\circ,C=22{}^\circ \text{ and }c\approx 5.4$.
For any triangle, The law of sines states that: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\operatorname{sinC}}$ The law of cosines states that: \begin{align} & {{a}^{2}}={{b}^{2}}+{{c}^{2}}-2\cdot b\cdot c\cdot \cos A \\ & {{b}^{2}}={{a}^{2}}+{{c}^{2}}-2\cdot a\cdot c\cdot \cos B \\ & {{c}^{2}}={{a}^{2}}+{{b}^{2}}-2\cdot a\cdot b\cdot \cos C \\ \end{align} Using the law of sines we get, \begin{align} & \frac{a}{\sin A}=\frac{b}{\sin B} \\ & \frac{12.4}{\sin A}=\frac{8.7}{\sin 37{}^\circ } \\ & 8.7\left( \sin A \right)=12.4\left( \sin 37{}^\circ \right) \\ & \sin A=\frac{12.4\left( \sin 37{}^\circ \right)}{8.7} \end{align} It can further be simplified to get a measure of angle C. \begin{align} & \sin A=0.85 \\ & A={{\sin }^{-1}}\left( 0.85 \right) \\ & A\approx 59{}^\circ,121{}^\circ \end{align} Now using the angle sum property for both values of A we will get the measure of angle C. For $A=59{}^\circ$, \begin{align} & 59{}^\circ +37{}^\circ +C=180{}^\circ \\ & C=180{}^\circ -96{}^\circ \\ & C=84{}^\circ \end{align} For $A=121{}^\circ$, \begin{align} & 121{}^\circ +37{}^\circ +C=180{}^\circ \\ & C=180{}^\circ -158{}^\circ \\ & C=22{}^\circ \end{align} Now using the law of sines we will obtain two values of side c for different angles C: For $C=84{}^\circ$ \begin{align} & \frac{8.7}{\sin 37{}^\circ }=\frac{c}{\sin 84{}^\circ } \\ & c\cdot \sin 37{}^\circ =8.7\left( \sin 84{}^\circ \right) \\ & c=\frac{8.7\left( \sin 84{}^\circ \right)}{\sin 37{}^\circ } \end{align} This gives $c\approx 14.4$. For $C=22{}^\circ$, \begin{align} & \frac{8.7}{\sin 37{}^\circ }=\frac{c}{\sin 22{}^\circ } \\ & c\cdot \sin 37{}^\circ =8.7\left( \sin 22{}^\circ \right) \\ & c=\frac{8.7\left( \sin 22{}^\circ \right)}{\sin 37{}^\circ } \end{align} This gives $c\approx 5.4$. So, \begin{align} & A=59{}^\circ,C=84{}^\circ \text{ and }c\approx 14.4 \\ & A=121{}^\circ,C=22{}^\circ \text{ and }c\approx 5.4 \\ \end{align} Hence, $A=59{}^\circ,C=84{}^\circ \text{ and }c\approx 14.4\text{ or }A=121{}^\circ,C=22{}^\circ \text{ and }c\approx 5.4$ are the two sets of possible values.