Precalculus (6th Edition) Blitzer

Published by Pearson
ISBN 10: 0-13446-914-3
ISBN 13: 978-0-13446-914-0

Chapter 6 - Section 6.7 - The Dot Product - Exercise Set - Page 799: 77


$5\left( \cos 55{}^\circ +i\sin 55{}^\circ \right),5\left( \cos 175{}^\circ +i\sin 175{}^\circ \right),5\left( \cos 295{}^\circ +i\sin 295{}^\circ \right)$

Work Step by Step

Method for finding complex roots: To find $n$ distinct complex roots for any complex number $z=r\left( \cos \theta +i\sin \theta \right)$, if $z\ne 0$, in radians we use the formula given below: ${{z}_{k}}=\sqrt[n]{r}\left[ \cos \left( \frac{\theta +360{}^\circ \cdot k}{n} \right)+i\sin \left( \frac{\theta +360{}^\circ \cdot k}{n} \right) \right]$ Where $k=0,1,2,3,....,n-1$. So, the cube roots of $125\left( \cos 165{}^\circ +i\sin 165{}^\circ \right)$ are ${{z}_{k}}=\sqrt[3]{125}\left[ \cos \left( \frac{165{}^\circ +360{}^\circ \cdot k}{3} \right)+i\sin \left( \frac{165{}^\circ +360{}^\circ \cdot k}{3} \right) \right],k=0,1,2$ Therefore, we can find the three complex cube roots in the following manner: $\begin{align} & {{z}_{0}}=\sqrt[3]{125}\left[ \cos \left( \frac{165{}^\circ +360{}^\circ \cdot 0}{3} \right)+i\sin \left( \frac{165{}^\circ +360{}^\circ \cdot 0}{3} \right) \right] \\ & =5\left( \cos \frac{165{}^\circ }{3}+i\sin \frac{165{}^\circ }{3} \right) \\ & =5\left( \cos 55{}^\circ +i\sin 55{}^\circ \right) \end{align}$ $\begin{align} & {{z}_{1}}=\sqrt[3]{125}\left[ \cos \left( \frac{165{}^\circ +360{}^\circ \cdot 1}{3} \right)+i\sin \left( \frac{165{}^\circ +360{}^\circ \cdot 1}{3} \right) \right] \\ & =5\left( \cos \frac{525{}^\circ }{3}+i\sin \frac{525{}^\circ }{3} \right) \\ & =5\left( \cos 175{}^\circ +i\sin 175{}^\circ \right) \end{align}$ $\begin{align} & {{z}_{2}}=\sqrt[3]{125}\left[ \cos \left( \frac{165{}^\circ +360{}^\circ \cdot 2}{3} \right)+i\sin \left( \frac{165{}^\circ +360{}^\circ \cdot 2}{3} \right) \right] \\ & =5\left( \cos \frac{885{}^\circ }{3}+i\sin \frac{885{}^\circ }{3} \right) \\ & =5\left( \cos 295{}^\circ +i\sin 295{}^\circ \right) \end{align}$ So, the cube roots are $5\left( \cos 55{}^\circ +i\sin 55{}^\circ \right),\text{ }5\left( \cos 175{}^\circ +i\sin 175{}^\circ \right),\text{ and }5\left( \cos 295{}^\circ +i\sin 295{}^\circ \right)$.
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