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Chapter 6: Problem 7

Write $$ \frac{1}{6\left(\cos \frac{\pi}{11}+i \sin \frac{\pi}{11}\right)} $$ in polar form.

Short Answer

Expert verified
The polar form of the given complex number is: \[ z = \frac{1}{6}\left(\cos\left(-\frac{\pi}{11}\right)+i\sin\left(-\frac{\pi}{11}\right)\right) \]

Step by step solution

01

Identify the given complex number

The given complex number is: \[ z = \frac{1}{6\left(\cos \frac{\pi}{11} + i \sin \frac{\pi}{11}\right)} \] We will now try to find the magnitude and angle of this complex number.
02

Find the magnitude

Magnitude \(r\) of a complex number in the form \(a(\cos(\theta) + i\sin(\theta))\) is given by \(|r|=\left|a\right|\). In our case, \[ |z| = \left|\frac{1}{6}\right| = \frac{1}{6} \]
03

Find the angle

The angle of the complex number in the form \(a(\cos(\theta) + i\sin(\theta))\) is given by \(\theta\). For our complex number, \[ \theta = - \frac{\pi}{11} \] We choose the negative of the angle because the given complex number is the reciprocal.
04

Represent the complex number in polar form

Now, we have the magnitude \(r = \frac{1}{6}\) and angle \(\theta = - \frac{\pi}{11}\). Therefore, the polar form of the given complex number is: \[ z = \frac{1}{6}\left(\cos\left(-\frac{\pi}{11}\right)+i\sin\left(-\frac{\pi}{11}\right)\right) \]

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