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Chapter 6: Problem 12

Plot each complex number. Then write the complex number in polar form. You may express the argument in degrees or radians. $$1+i \sqrt{3}$$

Short Answer

Expert verified
The complex number \(1+i\sqrt{3}\) in polar form is \(2(\cos60掳+i\sin60掳)\).

Step by step solution

01

- Identify the Complex Number

The given complex number is \(1+i \sqrt{3}\), which is expressed in rectangular form. Here, the real part is 1 and the imaginary part is \(\sqrt{3}\)
02

- Plot the Complex Number

The complex number \(1 + i\sqrt{3}\) corresponds to the point (1, \(\sqrt{3}\)) in the complex plane. Start from the origin, move one unit along the real(axis x) axis and then \(\sqrt{3}\) units along the imaginary (axis y) axis.
03

- Convert to Polar Form

To convert a complex number from rectangular form to polar form, you need to calculate the magnitude (r) and the argument (胃). The magnitude r is the distance from the origin to the point and is given by the formula \(r = \sqrt{(real \ part)^2 + (imaginary \ part)^2}\), so \(r = \sqrt{1^2 + (\sqrt{3})^2}=2\). The argument 胃 is the angle made with the positive real axis. We can find 胃 with the formula \(胃 = \arctan{\frac{imaginary \ part}{real \ part}}\), so \(胃 = \arctan{\frac{\sqrt{3}}{1}}=60掳\).
04

- Write in Polar Form

Finally, the complex number in polar form is written as \(r(cos胃+isin胃)\). So the polar form of our number is \(2(\cos60掳+i\sin60掳)\).

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Most popular questions from this chapter

Prove that the projection of \(\mathbf{v}\) onto \(\mathbf{i}\) is \((\mathbf{v} \cdot \mathbf{i}) \mathbf{i}\).

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