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 6.5 Sum and Difference Formulas - 6.5 Assess Your Understanding - Page 508: 29



Work Step by Step

Apply the difference formula: $\sin(A-B)=\sin(A)\cos(B) -\cos(A)\sin(B)$ We know that $\sin$, $\csc$, and $\tan$ are odd trigonometric functions. This implies that $f(-\theta)=-f(\theta)$ So, $\sin(-\theta)=-\sin(\theta)$ Therefore, $$\sin(20^\circ) \ \cos(80^\circ) -\cos(20^\circ) \ \sin(80^\circ)\\ =\sin \ (\dfrac{\pi}{12} -\dfrac{ 7 \pi}{12})\\ =\sin \ (\dfrac{ -6\pi}{12})\\ =\sin \ (\dfrac{ - \pi}{2}) \\=-1$$
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