Problem 3 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Express \((1-\cos \theta+i \sin \theta)\) in polar form.

Step-by-Step Solution

Answer

The polar form is \( 2 \sin \left( \frac{\theta}{2} \right) \left( \cos(\pi - \frac{\theta}{2}) + i \sin(\pi - \frac{\theta}{2}) \right) \).

Step 1: Understand the Formulas

To express a complex number in polar form, we need to find its magnitude and argument. The polar form is given by \[ r (\cos \phi + i \sin \phi) \]where \( r \) is the magnitude and \( \phi \) is the argument of the complex number.

Step 2: Identify the Real and Imaginary Parts

The given complex number is \((1 - \cos \theta) + i \sin \theta\). Here, the real part is \(1 - \cos \theta\) and the imaginary part is \(\sin \theta\).

Step 3: Calculate the Magnitude

The magnitude \( r \) of a complex number \(a + bi\) is given by \[ r = \sqrt{a^2 + b^2} \]For our complex number, \[ r = \sqrt{(1 - \cos \theta)^2 + (\sin \theta)^2} \]Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \), simplify this to:\[ r = \sqrt{1 - 2\cos \theta + \cos^2 \theta + \sin^2 \theta} = \sqrt{2 - 2\cos \theta} \]\[ r = \sqrt{2(1 - \cos \theta)} \]\[ r = 2 \sin \left( \frac{\theta}{2} \right) \] by using the half-angle identity \(1 - \cos \theta = 2\sin^2 \left( \frac{\theta}{2} \right)\).

Step 4: Calculate the Argument

The argument \( \phi \) is given by the angle whose cosine is \( \cos \phi = \frac{a}{r} \) and sine is \( \sin \phi = \frac{b}{r} \).For our complex number,\[ \tan \phi = \frac{\sin \theta}{1 - \cos \theta} \]Using the identity \( \tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta} \), the argument \( \phi = \pi - \frac{\theta}{2} \) for this quadrant.

Step 5: Express in Polar Form

Substitute the value of \( r \) and \( \phi \) into the polar form expression:\[ 2 \sin \left( \frac{\theta}{2} \right) \left( \cos(\pi - \frac{\theta}{2}) + i \sin(\pi - \frac{\theta}{2}) \right) \]

Key Concepts

Polar Form

Complex numbers can be represented in several ways. One of the most insightful representations is the polar form. Instead of the traditional rectangular format, where a complex number is written as \( a + bi \), we use the polar form:

• \( r(\cos \phi + i \sin \phi) \), where:
• \( r \) is the magnitude or modulus of the complex number.
• \( \phi \) is the angle, known as the argument.

This form is powerful because it aligns with the geometric interpretation of complex numbers on the coordinate plane.
You visualize \( r \) as the distance from the origin to the point, and \( \phi \) as the angle between the positive x-axis and the line connecting the origin to the point. Polar form is particularly useful in multiplying or dividing complex numbers as it simplifies the calculations.

Magnitude

The magnitude of a complex number is a crucial concept. It represents the "length" or "distance" of the complex number from the origin in the complex plane.
Mathematically, the magnitude \( r \) of a complex number \( a + bi \) can be found using the formula:

• \( r = \sqrt{a^2 + b^2} \)

The squared components \( a^2 \) and \( b^2 \) sum up to give the square of the magnitude.
For our exercise, the magnitude was simplified using trigonometric identities:

• \( r = \sqrt{2(1 - \cos \theta)} \)
• Translating to \( r = 2 \sin \left( \frac{\theta}{2} \right) \) using the half-angle identity.

Understanding how to find the magnitude is essential in transitioning from the rectangular to polar form.

Argument

The argument of a complex number is another critical attribute.
It describes the direction or angle of the vector representing the complex number, relative to the positive x-axis.

• The argument \( \phi \) is determined using the arc tangent of the imaginary part over the real part, giving us \( \tan \phi = \frac{b}{a} \).

For our specific complex number, we derived:

• \( \tan \phi = \frac{\sin \theta}{1 - \cos \theta} \)
• This leads to \( \phi = \pi - \frac{\theta}{2} \) in the relevant quadrant.

The argument provides insight into the orientation of the complex number in the plane, and is critical for converting to polar form, especially when considering different quadrants.

Half-Angle Identity

A half-angle identity is a trigonometric identity that allows us to express a trigonometric function using half of an angle.
These identities are incredibly useful in converting expressions and simplifying trigonometric equations, as seen in our exercise.

• One common identity is: \( 1 - \cos \theta = 2 \sin^2 \left( \frac{\theta}{2} \right) \)
• Similarly, \( 1 + \cos \theta = 2 \cos^2 \left( \frac{\theta}{2} \right) \)

These formulas were used to simplify the magnitude of our complex number:

• \( \sqrt{2(1 - \cos \theta)} \) becomes \( 2 \sin \left( \frac{\theta}{2} \right) \).

Employing half-angle identities helps in reducing expressions and finding elegant solutions in trigonometric calculations.