Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Chapter 6 - Analytic Trigonometry - Section 6.6 Double-angle and Half-angle Formulas - 6.6 Assess Your Understanding - Page 518: 26


$\dfrac{2}{\sqrt{{2 -\sqrt 2}}}$

Work Step by Step

The inverse Identity for secant can be expressed as: $\csc a =\dfrac {1}{\sin a}$ or, $\csc a =(\sin a)^{-1}$ Therefore, $\csc ({\dfrac{7 \pi }{8}})=( \sqrt{\dfrac{1 - \cos ({\dfrac{ 7 \pi }{4}})}{2}} )^{-1} \\=[ \sqrt{\dfrac{1 -\dfrac{\sqrt 2}{2}}{2}}]^{-1} \\=[\dfrac{\sqrt {2 -\sqrt 2}}{2}]^{-1}\\ = \dfrac{2}{\sqrt{{2 -\sqrt 2}}}$
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