Precalculus (6th Edition) Blitzer

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

Chapter 4 - Section 4.2 - Trigonometric Functions: The Unit Circle - Exercise Set - Page 550: 110


The simplified form of the expression ${{\left( \frac{a}{c} \right)}^{2}}+{{\left( \frac{b}{c} \right)}^{2}}$ is $1$.

Work Step by Step

Consider the expression, ${{\left( \frac{a}{c} \right)}^{2}}+{{\left( \frac{b}{c} \right)}^{2}}$ $\begin{align} & {{\left( \frac{a}{c} \right)}^{2}}+{{\left( \frac{b}{c} \right)}^{2}}=\frac{{{a}^{2}}}{{{c}^{2}}}+\frac{{{b}^{2}}}{{{c}^{2}}} \\ & =\frac{{{a}^{2}}+{{b}^{2}}}{{{c}^{2}}} \end{align}$ The Pythagorean identity is ${{c}^{2}}={{a}^{2}}+{{b}^{2}}$. Substitute ${{c}^{2}}={{a}^{2}}+{{b}^{2}}$ in the expression $\frac{{{a}^{2}}+{{b}^{2}}}{{{c}^{2}}}$. $\begin{align} & {{\left( \frac{a}{c} \right)}^{2}}+{{\left( \frac{b}{c} \right)}^{2}}=\frac{{{c}^{2}}}{{{c}^{2}}} \\ & =1 \end{align}$ Therefore, the simplified form of the expression ${{\left( \frac{a}{c} \right)}^{2}}+{{\left( \frac{b}{c} \right)}^{2}}$ is $1$.
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