91Ó°ÊÓ

Actual Speed and True Course. An airplane is flying on a course of \(285^{\circ}\) as measured from due north at \(300 \mathrm{mph}\). The wind is blowing due south at \(30 \mathrm{mph}\). Represent their respective vectors as complex numbers written in polar form, and determine the resultant speed and direction vector.

In Exercises 79-82, explain the mistake that is made. Find \((\sqrt{2}+i \sqrt{2})^{6}\) Solution: Raise each term to the sixth power. \((\sqrt{2})^{6}+i^{6}(\sqrt{2})^{6}\) Simplify. Let \(i^{6}=i^{4} \cdot i^{2}=-1\). \(8+8 i^{6}\) Let \(i^{6}=i^{4} \cdot i^{2}=-1 . \quad 8-8=0\) This is incorrect. What mistake was made?

In Exercises 81 and 82 , explain the mistake that is made. Express \(z=-3+8 i\) in polar form. Solution: Find \(r\). Find \(\theta\). $$ r=\sqrt{x^{2}+y^{2}}=\sqrt{9+64}=\sqrt{73} $$ $$ \tan \theta=-\frac{8}{3} $$ $$ \theta=\tan ^{-1}\left(-\frac{8}{3}\right) \approx-69.44^{\circ} $$ Write the complex number in polar form. $$ z \approx \sqrt{73}\left[\cos \left(-69.44^{\circ}\right)+i \sin \left(-69.44^{\circ}\right)\right] $$ This is incorrect. What mistake was made?

Convert \((-a, b)\) to polar coordinates. Assume \(a>0\) and \(b>0\).

Find the modulus of \(z=a\), where \(a\) is a negative real number.

Given \(r=2 \cos \left(\frac{3 \theta}{2}\right)\), find the \(\theta\)-intervals for the petal in the first quadrant.

For Exercises 99 and 100, use graphing calculators to convert complex numbers from rectangular to polar form. Use the Abs command to find the modulus and the Angle command to find the angle. Find abs \((1+i)\). Find angle \((1+i)\). Write \(1+i\) in polar form.