Problem 42 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Let \(z\) be a complex number with modulus 2 and argument \(\frac{2 \pi}{3}\), then \(z\) is equal to (A) \(-1+i \sqrt{3}\) (B) \(1-i \sqrt{3}\) (C) \(-\frac{1}{2}+\frac{i \sqrt{3}}{2}\) (D) None of these

Step-by-Step Solution

Answer

The correct answer is (A) \(-1 + i \sqrt{3}\).

Step 1: Recall the polar form formula

A complex number in polar form is represented as \( z = r(\cos \theta + i \sin \theta) \), where \( r \) is the modulus and \( \theta \) is the argument.

Step 2: Apply modulus and argument

For this problem, the modulus \( r = 2 \) and the argument \( \theta = \frac{2\pi}{3} \). Plug these values into the polar form formula: \( z = 2(\cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3}) \).

Step 3: Calculate cosine and sine

The cosine of \( \frac{2\pi}{3} \) is \(-\frac{1}{2}\), and the sine of \( \frac{2\pi}{3} \) is \(\frac{\sqrt{3}}{2}\).

Step 4: Substitute and simplify

Substitute the values from Step 3 into the equation from Step 2: \( z = 2(-\frac{1}{2} + i \frac{\sqrt{3}}{2}) \). Simplify this to get \( z = -1 + i\sqrt{3} \).

Step 5: Compare with options

The simplified expression for \( z \) is \(-1 + i \sqrt{3}\), which matches option (A).

Key Concepts

Polar Form of Complex Numbers

Understanding the polar form of complex numbers is crucial for representing complex numbers in terms of their magnitude and directional angle. The polar form is written as \( z = r(\cos \theta + i \sin \theta) \), where \( r \) is the modulus or magnitude of the complex number, and \( \theta \) is the argument or angle.

• Modulus (\( r \)): Represents the distance of the complex number from the origin in the complex plane.
• Argument (\( \theta \)): The angle \( \theta \) is measured from the positive x-axis counter-clockwise to the line representing the complex number.

The polar form is useful because it simplifies multiplication and division of complex numbers. When two complex numbers are multiplied, their moduli are multiplied and their arguments are added. Similarly, for division, the moduli are divided, and the arguments are subtracted. Keeping these operations in mind can be particularly useful when dealing with complex numbers in various branches of mathematics and engineering.

Modulus and Argument

The modulus and argument are key components in understanding complex numbers in polar form. Let's delve deeper into each component:

• Modulus: Given a complex number \( z = a + bi \), the modulus \( r \) is calculated as \( r = \sqrt{a^2 + b^2} \). It represents the length or magnitude of the vector formed by the complex number in the complex plane.


• Argument: The argument of a complex number is the direction angle \( \theta \) formed with the positive x-axis. It is calculated using the tangent function, \( \theta = \tan^{-1}(\frac{b}{a}) \). Depending on the quadrant where the complex number lies, adjustments may be necessary to find the correct angle.

For our example with \( r = 2 \) and \( \theta = \frac{2\pi}{3} \), the modulus tells us how far \( z \) is from the origin, while the argument provides the direction. Together, they provide a complete description of the complex number in its polar form.

Trigonometric Representation of Complex Numbers

The trigonometric representation of complex numbers is another way of expressing them using trigonometric functions. This representation stems directly from the polar form \( z = r(\cos \theta + i \sin \theta) \).
In this form, the real part of the complex number is \( r \cos \theta \), and the imaginary part is \( r \sin \theta \). This directly relates the complex number to the unit circle.
Key Points:

• Real and Imaginary Parts: The real part is found by multiplying the modulus by the cosine of the argument, while the imaginary part is found by multiplying the modulus by the sine of the argument.
• Ease of Use: The trigonometric representation is particularly useful when performing rotations and transformations on complex numbers, as well as simplifying complex number operations.

This approach also highlights the interconnectedness of complex numbers with trigonometry, whereby angles (arguments) and distances (moduli) are foundational to understanding them in a spatial manner.