Problem 153 - A Complete Resource Book in Mathematics for JEE Main

Question

The centre of a regular polygon of \(n\) sides is located at the point \(z=0\), and one of its vertex \(z_{1}\) is known. If \(z_{2}\) be the vertex adjacent to \(z_{1}\), then \(z_{2}\) is equal to (A) \(z_{1}\left(\cos \frac{2 \pi}{n}+i \sin \frac{2 \pi}{n}\right)\) (B) \(z_{1}\left(\cos \frac{\pi}{n}+i \sin \frac{\pi}{n}\right)\) (C) \(z_{1}\left(\cos \frac{2 \pi}{n}-i \sin \frac{2 \pi}{n}\right)\) (D) \(z_{1}\left(\cos \frac{\pi}{n}-i \sin \frac{\pi}{n}\right)\)

Step-by-Step Solution

Answer

(A) \(z_{1}\left(\cos \frac{2 \pi}{n}+i \sin \frac{2 \pi}{n}\right)\).

Step 1: Understand the Problem

We are given that a regular polygon with \(n\) sides is centered at the origin \(z = 0\), and we have one vertex \(z_1\). We need to find the adjacent vertex \(z_2\). The vertices of a regular polygon are evenly spaced on a circle.

Step 2: Understand the Rotation

Since we are dealing with a regular polygon centered at the origin, moving from one vertex to the adjacent one implies rotating by a certain angle about the origin. This rotation angle is the central angle \( \theta = \frac{2 \pi}{n} \) radians, where \( n \) is the number of sides.

Step 3: Apply Complex Number Rotation

In complex numbers, rotation by angle \( \theta \) can be represented as multiplication by \( e^{i\theta} \), where \( \theta = \frac{2\pi}{n} \). Therefore, rotating \( z_1 \) by this angle to find \( z_2 \) gives: \( z_2 = z_1 \cdot e^{i \frac{2\pi}{n}} \).

Step 4: Express Rotation in Trigonometric Form

The expression \( e^{i \frac{2\pi}{n}} \) can be rewritten using Euler's formula: \( e^{i \theta} = \cos \theta + i \sin \theta \). Therefore, \( e^{i \frac{2\pi}{n}} = \cos \frac{2\pi}{n} + i \sin \frac{2\pi}{n} \).

Step 5: Find the Correct Answer

Substitute the trigonometric form into the expression for \( z_2 \): \( z_2 = z_1 (\cos \frac{2\pi}{n} + i \sin \frac{2\pi}{n}) \). This corresponds to option (A).

Key Concepts

Regular Polygon

A regular polygon is a geometric shape with all sides and angles equal. These shapes can be visualized as polygons with symmetrically distributed vertices. Here is what you should know about regular polygons in geometry:

• They appear frequently in geometry and real-life applications such as tiling patterns, crystals, and architecture.
• The number of sides, denoted by \( n \), determines the central angle between consecutive vertices.
• When plotted on the complex plane, each vertex of the polygon corresponds to a complex number which can be interpreted as a point in 2D space.

Understanding regular polygons is fundamental when dealing with complex numbers in geometry, particularly when they are centered at the origin of a plane as this makes calculations more straightforward.

Complex Number Rotation

Rotation in the complex plane is a vital concept where you rotate one complex number about another. Here’s how it works:

• Consider a complex number represented as \( z = x + iy \), where \( x \) and \( y \) are the real and imaginary parts, respectively.
• To rotate a complex number around the origin, we multiply it by another complex number corresponding to the rotation angle \( \theta \). This is done using the expression \( e^{i\theta} \), which encapsulates the rotation.
• This type of rotation is insightful when working with regular polygons. When you rotate one vertex of a regular polygon to the next, you are essentially performing a complex number rotation.

Rotations are at the core of discussions involving regular polygons and their vertex arrangements in complex geometry.

Euler's Formula

Euler's formula is one of the most celebrated equations in mathematics. It builds a beautiful bridge between exponential functions and trigonometry. Here’s what it states and how it applies:

• The formula is written as \( e^{i\theta} = \cos \theta + i \sin \theta \).
• It is incredibly useful for converting the radius-angle form in polar coordinates to a more familiar form using sine and cosine. Hence its significance when working with complex number rotations.
• For regular polygons, Euler's formula allows us to express each vertex as a rotation of another, giving a clear trigonometric understanding of their arrangement in the complex plane.

By utilizing Euler's formula, one can easily switch between different ways of expressing complex numbers, making it a powerful tool in both pure and applied mathematics.

Central Angle

The central angle is the angle subtended by each side of the polygon at its center. It's crucial when working with regular polygons:

• The measure of the central angle \( \theta \) in radians for a regular polygon with \( n \) sides is given by \( \theta = \frac{2\pi}{n} \).
• This angle defines how far apart the vertices are spaced around the circle that circumscribes the polygon.
• In the context of complex numbers, it helps us determine the necessary rotation angle to move from one vertex to the next along the circle.

Central angles hold great significance in making geometric arguments about symmetry and regularity of shapes in planar geometry.

Trigonometric Form

The trigonometric form of complex numbers provides a way to express them using trigonometric functions. Key aspects include:

• Any complex number can be written in the form \( z = r(\cos \theta + i\sin \theta) \), where \( r \) is the modulus and \( \theta \) the argument or angle made with the positive real axis.
• This form is particularly useful in problems involving rotation and periodicity, such as when determining the positions of the vertices of a regular polygon.
• The trigonometric form also simplifies the multiplication of complex numbers, especially when discussing rotations, as seen by \( r_1 e^{i\theta_1} \cdot r_2 e^{i\theta_2} = (r_1 r_2) e^{i(\theta_1 + \theta_2)} \).

The trigonometric form of complex numbers is indispensable for elegantly handling rotations and other transformations in the complex plane.