## Precalculus (6th Edition) Blitzer

Published by Pearson

# Chapter 6 - Review Exercises - Page 798: 4

#### Answer

The values $A=55{}^\circ,B=27{}^\circ \text{ and }C=98{}^\circ \text{ or }A=55{}^\circ,B=45{}^\circ \text{ and }C=80{}^\circ$.

#### Work Step by Step

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} From the law of cosines: \begin{align} & {{a}^{2}}={{b}^{2}}+{{c}^{2}}-2\cdot b\cdot c\cdot \cos A \\ & {{117}^{2}}={{66}^{2}}+{{142}^{2}}-2\cdot 66\cdot 142\cdot \cos A \\ & \cos A=\frac{\left( {{66}^{2}}+{{142}^{2}} \right)-{{117}^{2}}}{2\cdot 66\cdot 142} \\ & \cos A=\frac{24520-13689}{18744} \end{align} We will simplify it further \begin{align} & \cos A=0.57 \\ & A={{\cos }^{-1}}\left( 0.57 \right) \\ & A\approx 55{}^\circ \end{align} Using the law of sines we get, \begin{align} & \frac{a}{\sin A}=\frac{c}{\sin C} \\ & \frac{117}{\sin 55{}^\circ }=\frac{142}{\sin C} \\ & 117\cdot \sin C=142\cdot \sin 55{}^\circ \\ & \sin C=\frac{142\cdot \sin 55{}^\circ }{117} \end{align} Proceed further to obtain the measure of angle C \begin{align} & \sin C=0.99 \\ & C={{\sin }^{-1}}\left( 0.99 \right) \\ & C\approx 98{}^\circ,80{}^\circ \end{align} Using the angle sum property for both values of angle C. For $C=98{}^\circ$, \begin{align} & 55{}^\circ +B+98{}^\circ =180{}^\circ \\ & B=180{}^\circ -153{}^\circ \\ & B=27{}^\circ \end{align} For $C=80{}^\circ$, \begin{align} & 55{}^\circ +B+80{}^\circ =180{}^\circ \\ & B=180{}^\circ -135{}^\circ \\ & B=45{}^\circ \end{align} So, the two combinations for the triangles are: \begin{align} & A=55{}^\circ,B=27{}^\circ \text{ and }C=98{}^\circ \\ & A=55{}^\circ,B=45{}^\circ \text{ and }C=80{}^\circ \\ \end{align} Hence, $A=55{}^\circ,B=27{}^\circ \text{ and }C=98{}^\circ \text{ or }A=55{}^\circ,B=45{}^\circ \text{ and }C=80{}^\circ$.

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