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 5.4 Sum and Difference Identities for Sine and Tangent - 5.4 Exercises - Page 227: 43



Work Step by Step

$$X=\sin(270^\circ-\theta)$$ According to sine difference identity: $$\sin(A-B)=\sin A\cos B-\cos A\sin B$$ Expand $X$: $$X=\sin270^\circ\cos\theta-\cos270^\circ\sin\theta$$ *About $\sin270^\circ$ and $\cos270^\circ$ $$\sin270^\circ=\sin(-90^\circ)\hspace{2cm}\cos270^\circ=\cos(-90^\circ)$$ From the Negative-Angle Identities: $$\sin(-\theta)=-\sin\theta\hspace{2cm}\cos(-\theta)=\cos\theta$$ So, $$\sin270^\circ=-\sin90^\circ=-1\hspace{2cm}\cos270^\circ=\cos90^\circ=0$$ We replace the values just being found back into $X$: $$X=(-1)\cos\theta-0\sin\theta$$ $$X=-\cos\theta$$ Overall, $$\sin(270^\circ-\theta)=-\cos\theta$$
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