Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 5 - Trigonometric Identities - Section 5.2 Verifying Trigonometric Identities - 5.2 Exercises - Page 210: 93

Answer

$\dfrac{2+5\cos{x}}{\sin{x}}=2\csc{x}+5\cot{x}$ is an identity

Work Step by Step

Use a graphing utility to graph $y=\dfrac{2+5\cos{x}}{\sin{x}}$ and $y=2\csc{x}+5\cot{x}$. (Refer to the graphs below.) Note that the graphs are exactly the same as all the curves coincide. This means that the given equation is an identity. RECALL: (1) $\csc{x}=\dfrac{1}{\sin{x}}$ (2) $\cot{x}=\dfrac{\cos{x}}{\sin{x}}$ Use the definitions above to obtain: \begin{align*} \dfrac{2+5\cos{x}}{\sin{x}}&=\frac{2}{\sin{x}}+\frac{5\cos{x}}{\sin{x}}\\\\ &=2\left(\frac{1}{\sin{x}}\right)+5\left(\frac{\cos{x}}{\sin{x}}\right)\\\\ &=2\csc{x}+5\cot{x} \end{align*} Therefore, $$\dfrac{2+5\cos{x}}{\sin{x}}=2\csc{x}+5\cot{x}$$
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