Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 6 - Section 6.2 - The Law of Cosines - Concept and Vocabulary Check - Page 729: 1


If $A$, $B$ and $C$ are the measures of the angle of a triangle, and $a$, $b$ and $c$ are the lengths of the sides opposite to these angles, then the Law of Cosines states that ${{a}^{2}}={{b}^{2}}+{{c}^{2}}-2bc\cos A$.

Work Step by Step

Consider the triangles given below and use the basic trigonometric ratios to obtain: From the figure, we get $a=c\cos B+b\cos C$ Similarly we get $\begin{align} & b=c\cos A+a\cos C \\ & c=a\cos B+b\cos A \\ \end{align}$ Therefore, $\begin{align} & {{a}^{2}}=ac\cos B+ab\operatorname{cosC} \\ & {{b}^{2}}=bc\cos A+ab\cos C \\ & {{c}^{2}}=ac\cos B+bc\cos A \\ \end{align}$ So, $\begin{align} & {{b}^{2}}+{{c}^{2}}-{{a}^{2}}=ab\cos C+bc\cos A+bc\cos A+ac\cos B-ac\cos B-ab\cos C \\ & =2bc\cos A \end{align}$ Hence, ${{a}^{2}}={{b}^{2}}+{{c}^{2}}-2bc\cos A$.
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