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


$\dfrac{2+5\cos{x}}{\sin{x}}=2\csc{x}+5\cot{x}$ 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, $$\dfrac{2+5\cos{x}}{\sin{x}}=2\csc{x}+5\cot{x}$$
Small 1554442670
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.