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Chapter 6: Problem 34

Write each complex number in rectangular form. If necessary, round to the nearest tenth. $$7\left(\cos \frac{3 \pi}{2}+i \sin \frac{3 \pi}{2}\right)$$

Short Answer

Expert verified
The rectangular form of the given complex number is -7i.

Step by step solution

01

Identify the magnitude and angle

In the polar form of the complex number \(7(\cos \frac{3 \pi}{2} +i \sin \frac{3 \pi}{2})\), the magnitude (r) is 7 and the angle (胃) is \(\frac{3 \pi}{2}\).
02

Compute the real and imaginary parts

The real and imaginary parts of the complex number can be calculated using the formulas \(x = r \cos 胃\) and \(y = r \sin 胃\). For this problem, \(x = 7 \cos \frac{3 \pi}{2} = 0\) and \(y = 7 \sin \frac{3 \pi}{2} = -7\).
03

Write the complex number in rectangular form

Replace the values of \(x\) and \(y\) to write the complex number in rectangular form: \(x + yi = 0 + (-7)i = -7i\).

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