Problem 203 - A Complete Resource Book in Mathematics for JEE Main 2017

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) \(\vec{i}\) (B) 1 (C) \(-1\) (D) \(-i\)

Step-by-Step Solution

Answer

The value of the expression is \(-i\) (option D).

Step 1: Recognize the Formula

Understand the expression given in the problem. The expression \(\sin \frac{2k\pi}{11} + i \cos \frac{2k\pi}{11}\) is similar to a complex number in the form \(a + ib\). Recall the Euler's formula, which states \(e^{ix} = \cos x + i\sin x\).

Step 2: Re-express Using Euler's Formula

Re-express \(\sin \frac{2k\pi}{11} + i \cos \frac{2k\pi}{11}\) using Euler's formula rewritten in a similar form: the expression is essentially \(i(e^{-i\frac{2k\pi}{11}})\) by substituting the expressions: \(\sin \theta = \cos(\frac{\pi}{2} - \theta)\) and \(\cos \theta = \sin(\frac{\pi}{2} - \theta)\).

Step 3: Recognize Sum as Part of a Root of Unity

The form \(e^{i\theta}= \cos\theta + i\sin\theta\) suggests that these are elements on the unit circle in the complex plane. Specifically, \(e^{-i\frac{2k\pi}{11}}\) represents the 11th roots of unity, rotated by \(-i\).

Step 4: Sum of the Roots of Unity

The characteristic property of roots of unity is that their sum over one full cycle is zero. Thus, the sum \(\sum_{k=1}^{10} e^{-i\frac{2k\pi}{11}}\) equals \(-1\) because it covers all roots except the one at \(k=11\) which is the unit root \(1\).

Step 5: Conclude the Result

Recognize that the factor \(i\) is multiplying the sum, so the result of \(i \cdot (-1) = -i\). This confirms that the sum of the original expression is \(-i\).

Key Concepts

Euler's Formula

Euler's formula is an essential concept in complex number theory, particularly noted for its beauty and widespread application. At its heart, Euler's formula connects trigonometry and complex exponentials. The formula is stated as: \[ e^{ix} = \cos x + i\sin x \]
This means you can represent complex numbers in exponential form, which simplifies many mathematical operations. The formula is used to convert between polar and rectangular coordinates in the complex plane.
In the context of our exercise, we utilize a variation of Euler's formula: \[ e^{-ix} = \cos x - i\sin x \] This helps us rewrite trigonometric expressions as exponential functions, making the computation of sums involving roots of unity much more straightforward.

• Euler's formula is crucial in analyzing periodic functions.
• It allows us to transform rotational movements in the complex plane succinctly.
• The formula simplifies calculus and differential equations by using exponential functions compared to trigonometric functions.

Understanding Euler's formula makes it easier to tackle various problems involving complex numbers and rotations in the complex plane.

Roots of Unity

Roots of unity are key elements in the study of complex numbers and are closely tied with Euler's formula. When solving polynomial equations of the form \( x^n = 1 \), the solutions are known as the nth roots of unity. These roots have several interesting properties:

• They are evenly spaced points on the unit circle in the complex plane.
• The principal nth root of unity is given by \( e^{2\pi i/n} \).
• The sum of all nth roots of unity is zero, simplifying computations like those in our exercise.

In our problem, we identified these elements when recognizing the expression \[ e^{-i\frac{2k\pi}{11}} \] as involving the 11th roots of unity. Notice how the factor of \(-i\) suggests a rotational transformation is applied.
The deep connection with roots of unity makes it simpler to determine results when dealing with sums over these cyclic elements, as their collective contributions often cancel out, leading to elegant and predictable results.

Complex Plane

The complex plane is a foundational concept in understanding complex numbers, providing a geometric interpretation of complex number arithmetic. Imagine the plane like a coordinate grid, but instead of \( (x, y) \), it uses \( (a, b) \) for the complex number \( a + ib \).

• Horizontal axis (real axis) represents the real part \(a\).
• Vertical axis (imaginary axis) represents the imaginary part \(b\).
• Each point on this plane corresponds to a unique complex number.

This plane allows us to visualize operations involving complex numbers, such as addition, multiplication, and importantly, rotation.
In our particular exercise, the complex plane is fundamental since we are considering points (roots of unity) that lie on the unit circle—a circle with a radius of one centered at the origin.
This circle represents all complex numbers that have absolute value 1, meaning any number \( e^{i\theta} \) is located on it. By understanding the complex plane, Euler's Formula, and roots of unity, we unlock a deeper comprehension of not just the specific problem presented but of complex numbers in general and their fascinating properties.