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Chapter 6: Problem 32

Write each complex number in rectangular form. If necessary, round to the nearest tenth. $$4\left(\cos \frac{5 \pi}{6}+i \sin \frac{5 \pi}{6}\right)$$

Short Answer

Expert verified
The given complex number in rectangular form is \(-2 + 3.5i\)

Step by step solution

01

Identify the modulus and argument

The given complex number is in the form \(r(\cos \theta + i\sin \theta)\). Here the modulus \(r = 4\) and the argument \(\theta = \frac{5 \pi}{6}\)
02

Calculate the real part of the complex number

The real part of the complex number \(a\) can be found using \(a = r\cos \theta\). Substituting the given values, we have \(a = 4\cos(\frac{5 \pi}{6}) = -2\)
03

Calculate the imaginary part of the complex number

The imaginary part of the complex number \(b\) can be found using \(b = r\sin \theta\). Substituting the given values, we have \(b = 4\sin(\frac{5 \pi}{6}) = 3.5\)
04

Write the complex number in rectangular form

With \(a = -2\) and \(b = 3.5\), the given complex number in rectangular form is \(-2 + 3.5i\)

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