Question

Express the given complex number in the form \(R(\cos \theta+\) \(i \sin \theta)=R e^{i \theta}\) $$ -1+\sqrt{3} i $$

Short answer

Question: Express the complex number \(-1 + \sqrt{3} i\) in polar form \(R(\cos \theta + i \sin \theta) = R e^{i \theta}\). Answer: \(2(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}) = 2e^{i\frac{2\pi}{3}}\)

Step-by-step solution

Identify the real and imaginary parts of the complex number

The given complex number is \(-1 + \sqrt{3} i\). Here, the real part is \(-1\), and the imaginary part is \(\sqrt{3}\).

Calculate the magnitude (R)

To find the magnitude of the complex number, apply the Pythagorean theorem using the real and imaginary parts: R = \(\sqrt{(-1)^2 + (\sqrt{3})^2}\) = \(\sqrt{1 + 3}\) = \(\sqrt{4}\) = 2.

Calculate the angle (θ)

To find the angle θ of the complex number, use the following formula: θ = arctan \(\frac{Imaginary\ Part}{Real\ Part}\) = arctan\(\frac{\sqrt{3}}{-1}\). Since the complex number lies in the second quadrant (negative real part and positive imaginary part), add π to the resulting angle: θ = arctan\(\frac{\sqrt{3}}{-1}\) + π = -\(\frac{\pi}{3}+\pi= \frac{2\pi}{3}\).

Write the complex number in polar form

Now that we have found the magnitude, R = 2, and the angle, θ = \(\frac{2\pi}{3}\), let's write the given complex number in the form \(R(\cos \theta + i \sin \theta) = R e^{i \theta}\): 2(\(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}\)) = 2\(e^{i\frac{2\pi}{3}}\) Thus, the given complex number in the desired form is : $$ 2(\cos \frac{2\pi}{3}+i \sin \frac{2\pi}{3})=2e^{i\frac{2\pi}{3}} $$

Key concepts

Magnitude of Complex Numbers

Understanding the concept of the magnitude of complex numbers is much like finding the length of a vector in two-dimensional space. It is often described as the distance from the origin (0,0) to the point represented by the complex number on the complex plane. This magnitude is denoted as |z| for a complex number z.

The mathematical formula to calculate the magnitude (or modulus) of a complex number represented as a + bi is: \[|z| = \sqrt{a^2 + b^2}\]
where 'a' is the real part and 'b' is the imaginary part of the complex number. Following this formula, for a complex number \(-1 + \sqrt{3}i\), we identify 'a' as -1 and 'b' as \(\sqrt{3}\). Plugging these values into the formula gives us a magnitude of 2, which, geometrically speaking, means the point is two units away from the origin in the complex plane.

Argument of Complex Numbers

The argument of a complex number corresponds to the angle it makes with the positive real axis, which is akin to the direction from the origin to the point representing the complex number in the polar coordinate system. The argument is often represented by the symbol \(\theta\) or arg(z).

The formula to calculate the argument for a complex number whose real part is 'a' and imaginary part is 'b' is: \[\theta = \arctan(\frac{b}{a})\]

In cases where the complex number is located in any quadrant other than the first, adjustments must be made to ensure the correct angle is found. In our exercise with the complex number \(-1 + \sqrt{3}i\), the real part is negative and the imaginary part is positive, so the number is in the second quadrant. This is why we added \(\pi\) to the calculated angle, resulting in an argument of \(\frac{2\pi}{3}\), which correctly places the angle in the second quadrant.

Euler's Formula

Euler's formula is a significant finding in mathematics that provides a powerful connection between complex numbers and trigonometry. It is given by:\[e^{i\theta} = \cos(\theta) + i\sin(\theta)\]
where 'e' is the base of the natural logarithm, 'i' is the imaginary unit, and \(\theta\) is the angle in radians.

This formula is especially useful when expressing complex numbers in polar form. The polar form of a complex number uses the magnitude and argument to show how far and in what direction to move from the origin to reach the number. If we have a magnitude 'R' and an argument \(\theta\), we use Euler's formula to write the complex number as \(Re^{i\theta}\).
Applying Euler's formula to our previous example, with 'R' being 2 and \(\theta\) as \(\frac{2\pi}{3}\), we can express the complex number \(-1 + \sqrt{3}i\) in polar form as \(2e^{i\frac{2\pi}{3}}\), succinctly capturing both its magnitude and direction in a single elegant notation.