Trigonometry (10th Edition)

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

Chapter 8 - Complex Numbers, Polar Equations, and Parametric Equations - Section 8.3 The Product and Quotient Theorems - 8.3 Exercises - Page 369: 12

Answer

$-\frac{21\sqrt 3}{2}-\frac{21}{2}i$

Work Step by Step

First, we use the product theorem to multiply the absolute values and add the arguments: $(3$ cis $300^{\circ})(7$ cis $270^{\circ})$ $=3(7)$ cis $(300^{\circ}+270^{\circ})$ $=21$ cis $(570^{\circ})$ Next, we change the expression into its equivalent form: $=21$ cis $(570^{\circ})$ $=21 (\cos 570^{\circ}+i\sin 570^{\circ})$ Since we know that $\cos 570^{\circ}=\cos 210^{\circ}=-\frac{\sqrt 3}{2}$ and $\sin 570^{\circ}=\sin 210^{\circ}=-\frac{1}{2}$, we subsitute these values in the expression and simplify: $21 (\cos 570^{\circ}+i\sin 570^{\circ})$ $=21 [-\frac{\sqrt 3}{2}+i(-\frac{1}{2})]$ $=-\frac{21\sqrt 3}{2}-\frac{21}{2}i$
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