Trigonometry (11th Edition) Clone

Published by Pearson
ISBN 10: 978-0-13-421743-7
ISBN 13: 978-0-13421-743-7

Chapter 8 - Review Exercises - Page 410: 41

Answer

The graphs of all the complex numbers will be lying on the line $y = - x$ in both the 4th quadrant with trigonometric form $x\sqrt{2}(cos315^\circ + isin315^\circ)$ and in the 2nd quadrant with trigonometric form $x\sqrt{2}(cos135^\circ + isin135^\circ)$ respectively.

Work Step by Step

As $z = x + yi$, when the imaginary part of $z$ is the negative of the real part of $z$, $y = - x$ then. The complex number, $z = x + yi$, will become $z = x - xi$. When $x \gt 0$, $z$ is at $315^\circ$ with absolute value $\sqrt{x^2 + (-x)^2} = x\sqrt{2}$, all the complex numbers will be lying on the line $y = - x$ in the 4th quadrant with trigonometric form $x\sqrt{2}(cos315^\circ + isin315^\circ)$. However, when $x \lt 0$, $z$ is at $135^\circ$ with absolute value $\sqrt{(-x)^2 + x^2} = x\sqrt{2}$ same, all the complex numbers will still be lying on the line $y = - x$ but in the 2nd quadrant with trigonometric form $x\sqrt{2}(cos135^\circ + isin135^\circ)$.
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.