Trigonometry (10th Edition)

Published by Pearson
ISBN 10: 0321671775
ISBN 13: 978-0-32167-177-6

Chapter 5 - Trigonometric Identities - Section 5.4 Sum and Difference Identities for Sine and Tangent - 5.4 Exercises - Page 220: 7

Answer

2 formulas differ in the sign of $\cos A\sin B$. While in $\sin (A-B)$, the sign is negative, in $\sin (A+B)$, the sign is positive. 2 formulas both contain $\sin A\cos B$ and $\cos A\sin B$. The sign of $\sin A\cos B$ in both formulas is also positive.

Work Step by Step

- Formula for $\sin(A-B)$ $$\sin(A-B)=\sin A\cos B-\cos A\sin B$$ - Formula for $\sin (A+B)$ $$\sin(A+B)=\sin A\cos B+\cos A\sin B$$ As shown above, the content of each formula is the same. Both $\sin (A-B)$ and $\sin (A+B)$ contain $\sin A\cos B$ and $\cos A\sin B$. The sign of $\sin A\cos B$ in both cases are alike, which is positive. The only difference is the sign of $\cos A\sin B$. - In $\sin (A-B)$, the sign of $\cos A\sin B$ is negative. - In $\sin (A+B)$, the sign of $\cos A\sin B$ is positive.
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