Problem 7 - Advanced Engineering Mathematics

Question

Write the given complex number in polar form. \(-\sqrt{3}+i\)

Step-by-Step Solution

Answer

The polar form is \(2(\cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6})\).

Step 1: Identify Components of the Complex Number

The given complex number is \(-\sqrt{3} + i\). This can be expressed as A + Bi where A is the real part and B is the imaginary part.Here, A = -\sqrt{3} and B = 1.

Step 2: Calculate the Magnitude (r)

Calculate the magnitude of the complex number using the formula \(r = \sqrt{A^2 + B^2}\).Substitute A and B to get \(r = \sqrt{(-\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2\).

Step 3: Calculate the Argument (θ)

The argument \(θ\) of the complex number can be found using \(θ = \tan^{-1}\left(\frac{B}{A}\right)\).Here, \(θ = \tan^{-1}\left(\frac{1}{-\sqrt{3}}\right)\).Since the complex number is in the second quadrant, it implies \(θ = \pi - \frac{\pi}{6} = \frac{5\pi}{6}\).

Step 4: Write the Polar Form

The polar form of a complex number is written as \(r(\cos θ + i \sin θ)\).Substitute the obtained values to get the polar form as \(2(\cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6})\).

Key Concepts

Complex Numbers

Complex numbers might seem a little intimidating at first, but they are not that hard to understand. They are numbers that have a real part and an imaginary part in the form of \(A + Bi\). Here, \(A\) is the real part, and \(Bi\) is the imaginary part. The letter \(i\) is used to denote the imaginary unit, with the property that \(i^2 = -1\). The complex number in the given example is \(-\sqrt{3} + i\), where \(-\sqrt{3}\) is the real part and \(i\) is the imaginary part.

• Real Part: Just like real numbers, this is the part that we are familiar with from everyday mathematics.
• Imaginary Part: This involves the imaginary unit \(i\), and it allows for solutions to equations that don't have real solutions.

Understanding complex numbers is crucial because they simplify the process of solving mathematical problems, especially in engineering and physics.

Magnitude of Complex Numbers

The magnitude of a complex number, also called the modulus, is a way to determine its distance from the origin in the complex plane. It's like finding the hypotenuse of a right triangle formed by the real and imaginary parts. The formula for the magnitude \(r\) of a complex number \(A + Bi\) is \(r = \sqrt{A^2 + B^2}\).For instance, in the given problem, the complex number is \(-\sqrt{3} + i\). - Substitute \(A = -\sqrt{3}\) and \(B = 1\) into the formula: \[ r = \sqrt{(-\sqrt{3})^2 + (1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \]This means the magnitude of \(-\sqrt{3} + i\) is 2. The magnitude helps us visualize the size of the complex number directly in a geometric sense.

Argument of Complex Numbers

The argument of a complex number is the angle it makes with the positive real axis in the complex plane. It's an essential part to describe a complex number in polar form.To calculate the argument \(θ\), you use the formula:\[θ = \tan^{-1}\left(\frac{B}{A}\right)\]In this case, you have \(-\sqrt{3} + i\) with \(A = -\sqrt{3}\) and \(B = 1\).- The initial calculation: \[ θ = \tan^{-1}\left(\frac{1}{-\sqrt{3}}\right) = -\frac{\pi}{6} \]But the complex number actually lies in the second quadrant, implying:- To adjust for quadrant, the angle becomes: \[ θ = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \]The argument \(\frac{5\pi}{6}\) shows us the direction from the origin to the point \(-\sqrt{3}, i\). Knowing the argument facilities transforming complex numbers between rectangular and polar forms.

Trigonometric Form of Complex Numbers

Presenting complex numbers in trigonometric or polar form gives a more flexible and often simpler way to work with them, especially in multiplication and division.The trigonometric form uses the magnitude and argument:The standard polar form is:\[ r(\cos θ + i \sin θ)\]Using our previous findings:The magnitude \(r = 2\), and the argument \(θ = \frac{5\pi}{6}\), hence the polar form is:- Substitute into the form: \[ 2(\cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6}) \]

• Cosine: Represents the horizontal component.
• Sine: Represents the vertical component.

This way of expressing complex numbers is particularly useful in fields like electrical engineering, where calculations benefit from this format.