## Trigonometry (11th Edition) Clone

Published by Pearson

# Chapter 5 - Trigonometric Identities - Section 5.3 Sum and Difference Identities for Cosine - 5.3 Exercises - Page 221: 80

#### Answer

The reduction formula is $$\sin\theta$$

#### Work Step by Step

*Summary of the method: For a formula $f(Q\pm\theta)$ 1) See that $Q$ terminates on the $x$ or $y$ axis. If it terminates on the $x$ axis, go for Case 1. If it terminates on the $y$ axis, go for Case 2. 2) Case 1: - For a small positive value of $\theta$, determinate $Q\pm\theta$ lies in which quadrant. - If $f\gt0$, use a $+$ sign. If $f\lt0$, use a $-$ sign. - The reduced form will have that sign, $f$ the function and $\theta$ the angle. 3) Case 2: - For a small positive value of $\theta$, determinate $Q\pm\theta$ lies in which quadrant. - If $f\gt0$, use a $+$ sign. If $f\lt0$, use a $-$ sign. - The reduced form will have that sign, cofunction of$f$ as the function and $\theta$ the angle. $$\cos(270^\circ+\theta)$$ 1) $270^\circ$ terminates on the $y$ axis. We go for Case 2. 2) As $\theta$ is a very small positive value, which means $\theta\gt0$, $$270^\circ\lt(270^\circ+\theta)\lt360^\circ$$ So $270^\circ+\theta$ lies in quadrant IV. 3) Cosine is positive in quadrant IV. So we use a $+$ sign. 4) In case 2, we use the cofunction of given formula, which here is sine, combined with the positive sign proved above. Overall, the reduced form would be $$\sin\theta$$

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