University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 7 - Section 7.3 - Hyperbolic Functions - Exercises - Page 417: 1


$\tanh x=\dfrac{-3}{5}$ $sech x=\dfrac{4}{5}$; $csch x=\dfrac{4}{-3}$; $coth x=\dfrac{5}{-3}$

Work Step by Step

As we are given that $\sinh x=\dfrac{-3}{4}$ Need to find all the hyperbolic functions. Use identity: $\cosh^2 x-\sinh^2x=1$ or, $\cosh^2 x-(\dfrac{-3}{4})^2=1 \implies \cosh^2 x=\dfrac{25}{16}$ and $\cosh x= \dfrac{5}{4}$ Now, $\tanh x= \dfrac{\sin hx}{\cosh x}=\dfrac{\dfrac{-3}{4}}{\dfrac{5}{4}}=\dfrac{-3}{5}$ $sech x= \dfrac{1}{\cosh x}=\dfrac{1}{\dfrac{5}{4}}=\dfrac{4}{5}$; $csch x= \dfrac{1}{\sinh x}=\dfrac{1}{\dfrac{-3}{4}}=\dfrac{4}{-3}$; $coth x= \dfrac{1}{\tanh x}=\dfrac{1}{\dfrac{-3}{5}}=\dfrac{5}{-3}$
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