Problem 203 - A Complete Resource Book in Mathematics for JEE Main

Question

The value of \(\sum_{k=1}^{10}\left(\sin \frac{2 k \pi}{11}+i \cos \frac{2 k \pi}{11}\right)\) is (A) \(i\) (B) 1 (C) \(-1\) (D) \(-i\)

Step-by-Step Solution

Answer

The correct answer is (A) \(i\).

Step 1: Understanding the Summation

We are asked to evaluate the sum \( \sum_{k=1}^{10}\left(\sin \frac{2 k \pi}{11}+i \cos \frac{2 k \pi}{11}\right) \). The terms inside the sum resemble the expression for a complex exponential form and can be rewritten using Euler's formula.

Step 2: Apply Euler's Formula

Euler's formula states that \( e^{i\theta} = \cos\theta + i\sin\theta \). Notice that in the given expression, the real and imaginary parts seem flipped compared to this formula. However, observe:\[-i\sin\theta + \cos\theta = -i \left( \sin\theta + i \cos\theta \right) = e^{i(\pi/2 + \theta)} \]Here, \( \theta = \frac{2k\pi}{11} \), thus the term is equivalent to \( e^{i\left(\frac{\pi}{2} + \frac{2k\pi}{11}\right)} \).

Step 3: Transform the Expression

Our task simplifies to evaluating:\[ \sum_{k=1}^{10} e^{i\left(\frac{\pi}{2} + \frac{2k \pi}{11} \right)} \]This is a geometric series in the complex plane with the common ratio as a complex exponential. The sum of the series revolves around the unit circle, capturing one complete loop.

Step 4: Evaluate the Geometric Sum

The angles \( \frac{2k\pi}{11} \) for \( k=1, 2, \ldots, 10 \) form a complete set around the circle, effectively completing a full circle when joined with \( k=0 \). Therefore, the corresponding complex numbers form a closed path around the origin, making the sum zero. Our calculations however introduced a shift by \( \frac{\pi}{2} \), effectively shifting the whole path by 90 degrees.

Step 5: Simplify and Find the Result

Since the path is a closed loop around the origin offset by 90 degrees, this implies: \[ e^{i\frac{\pi}{2} } \sum_{k=0}^{10} e^{i\frac{2k\pi}{11} } = e^{i\frac{\pi}{2} } \cdot 0 = 0 \]Here, \( e^{i\frac{\pi}{2}} = i \). Therefore, the shifted sum around the unit circle giving \( i \) multiplied by 0 results in 0.

Key Concepts

Euler's Formula

Euler's formula is a powerful tool in mathematics that links exponentials and trigonometric functions. This formula is expressed as follows: \( e^{i\theta} = \cos\theta + i\sin\theta \). It means that any complex exponential can be represented as a combination of sine and cosine, bridging the rectangular and polar forms of complex numbers.
In the context of complex numbers, \( i \) is the imaginary unit satisfying \( i^2 = -1 \). By using this formula, we're able to convert problems involving trigonometric functions into more manageable exponential problems.
In our exercise, Euler's formula helps us identify that the summation expression can be transformed into an exponential form, thereby simplifying the calculation. This transformation aids not only in problem-solving but also in visualizing complex numbers on the unit circle.

Geometric Series

A geometric series is a sum of terms each multiplied by a fixed constant called the common ratio. In our exercise, this common ratio is a complex exponential expression.
The general formula for the sum of a geometric series is:

• If \( a \) is the first term and \( r \) is the common ratio, then the sum \( S \) of \( n \) terms is given by \( S = a \frac{1-r^n}{1-r} \)

When \( |r| < 1 \), the infinite series converges. In terms of complex numbers, the series involves rotating vectors in the complex plane.
In this problem, we understood the given summation as a geometric series with complex numbers. These numbers, when summed, essentially describe a complete loop in the complex plane, which informs the resulting simplification to zero when no extra phase shifts are applied.

Unit Circle

The unit circle is a crucial concept when it comes to complex numbers and trigonometry. It is a circle with radius one, centered at the origin of the complex plane. The unit circle helps visualize the relationship between trigonometric functions and complex exponentials.
When we talk about complex exponentials, we essentially describe points traversing the unit circle. For \( e^{i\theta} \), the angle \( \theta \) describes a point on the circle.
The exercise leverages the unit circle by interpreting the series sum around it. Each term represents a point on this path, moving in increments set by the geometric series, wrapping around the circle in a complete loop before the phase shift is applied.

Complex Exponentials

Complex exponentials are expressions in the form \( e^{ix} \), where \( x \) is a real number. These expressions play a significant role in transforming trigonometric identities into exponential forms.
In the original exercise, the expressions \( e^{i(\frac{\pi}{2} + \frac{2k\pi}{11})} \) appear, showing how complex exponentials provide both amplitude and phase information of a waveform.
They capture oscillations neatly, enabling easy manipulation for summation and other operations. When complex exponentials circle the unit circle completing a revolution, they sum to zero if not shifted, as they evenly distribute around all quadrants. The 90-degree phase offset adjusts the resulting direction.