Problem 30 - Algebra and Trigonometry

Question

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 1+\sqrt{3} i $$

Step-by-Step Solution

Answer

The polar form of \(1 + \sqrt{3}i\) is \[2\left(\cos \frac{\pi}{3} + i\sin \frac{\pi}{3}\right)\].

Step 1: Understand the Problem

We need to express the complex number \(1 + \sqrt{3}i\) in polar form, which requires finding the modulus \(r\) and the argument \(\theta\).

Step 2: Calculate the Modulus

The modulus \(r\) of a complex number \(a + bi\) is given by \(r = \sqrt{a^2 + b^2}\). Here, \(a = 1\) and \(b = \sqrt{3}\). Therefore, \(r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2\).

Step 3: Find the Argument \(\theta\)

The argument \(\theta\) is the angle the vector makes with the positive real axis, found using \(\tan \theta = \frac{b}{a}\). Thus, \(\tan \theta = \frac{\sqrt{3}}{1} = \sqrt{3}\). Since \(\tan \frac{\pi}{3} = \sqrt{3}\), \(\theta = \frac{\pi}{3}\).

Step 4: Write the Polar Form

With \(r = 2\) and \(\theta = \frac{\pi}{3}\), the polar form is given by \(r(\cos \theta + i\sin \theta)\). Hence, the polar form is \[2\left(\cos \frac{\pi}{3} + i\sin \frac{\pi}{3}\right)\].

Key Concepts

Complex Numbers

Complex numbers are numbers that consist of two parts: a real part and an imaginary part. They are often represented in the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part, with \(i\) being the imaginary unit defined by \(i^2 = -1\). For example, in the complex number \(1 + \sqrt{3}i\), \(1\) is the real part and \(\sqrt{3}i\) is the imaginary part.

• The real part \(a\) is the component that would exist without the imaginary unit.
• The imaginary part \(b\), when paired with \(i\), provides the second dimension that allows complex numbers to represent points in a plane.

Complex numbers can be visualized on the complex plane, a two-dimensional graph where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This representation aids in understanding operations with and the behavior of complex numbers. When we move to expressing complex numbers in polar form, this visual analogy continues to aid understanding.

Modulus of a Complex Number

The modulus of a complex number provides its 'size' or 'length' when viewed as a vector on the complex plane. For a complex number \(a + bi\), its modulus, denoted as \(r\) or sometimes \(|z|\), is calculated using the Pythagorean theorem: \[r = \sqrt{a^2 + b^2}\]
For the specific complex number \(1 + \sqrt{3}i\), the modulus calculation is as follows:

• Real part \(a = 1\)
• Imaginary part \(b = \sqrt{3}\)

By substituting these values into the formula, we find the modulus as \(r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{4} = 2\). This tells us that the vector representing our complex number extends 2 units from the origin in the complex plane. Understanding the modulus is key to converting a complex number to its polar form, as it defines the radius of the circle on which the number lies.

Argument of a Complex Number

The argument of a complex number, usually denoted as \(\theta\), is the angle that the vector of the complex number makes with the positive real axis in the complex plane. It gives the direction of the vector. To find this angle, we use the tangent function:\[\tan \theta = \frac{b}{a}\]
Here's how we find the argument for \(1 + \sqrt{3}i\):

• Substitute the imaginary part \(b = \sqrt{3}\) and real part \(a = 1\) into the equation.
• This gives \(\tan \theta = \frac{\sqrt{3}}{1} = \sqrt{3}\).

Knowing that \(\tan \frac{\pi}{3} = \sqrt{3}\), we can conclude that \(\theta = \frac{\pi}{3}\).
This angle is critical when expressing the complex number in polar form, as it completes the representation by defining not just how far out the vector extends (given by the modulus), but also in what direction that vector points.