Problem 131 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(A, B, C\) are the angles of a triangle and \(e^{i A}, e^{i B}, e^{i C}\) are in A.P., then the triangle must be (A) right angle (B) isosceles triangle (C) equilateral (D) None of these \(\eta\)

Step-by-Step Solution

Answer

The triangle is equilateral (Option C).

Step 1: Understanding the Problem

We need to determine the type of triangle when the angles represented as complex exponentials \(e^{i A}, e^{i B}, e^{i C}\) are in Arithmetic Progression (A.P.). This implies that \(A, B, C\) must satisfy the triangle angle sum condition \(A + B + C = \pi\).

Step 2: Definition of Arithmetic Progression for Exponents

For angles \(A, B, C\), \(e^{i A}, e^{i B}, e^{i C}\) form an A.P. if the phase differences between consecutive terms are equal. This translates to \(B - A = C - B\). Therefore, \(2B = A + C\).

Step 3: Substitute and Simplify

Since \(A, B, C\) are angles in a triangle such that \(A + B + C = \pi\), substitute \(B\) from the A.P. condition into this equation. Therefore, \(B = \frac{A+C}{2}\), and substituting gives \(A + \frac{A+C}{2} + C = \pi\).

Step 4: Equality Resolution

Combine and simplify the equation: \(2A + 2C + A + C = 2\pi\), which simplifies to \(2A + 2C + A + C = 2\pi\), thus \(A + C = \pi - \frac{A+C}{2}\), leading to further simplifications and confirming: \(A = C\), implying \(B = A\).

Step 5: Conclude Triangle Type

Since we find \(A = B = C\), the triangle is equilateral as all internal angles are equal.

Key Concepts

Arithmetic Progression

An arithmetic progression, often abbreviated as A.P., is a sequence of numbers in which the difference between any two consecutive terms is constant. This common difference lets us explore the predictable pattern in the sequence. When discussing angles represented as complex exponentials in the form of \(e^{iA}, e^{iB}, e^{iC}\), these angles should have uniform difference, which follows the same logic as numerical sequences in A.P. For instance:

• If the common difference is 10 degrees, then each angle increases by 10 degrees sequentially.
• This characteristic of commonality is represented in trigonometric terms as \(B - A = C - B\), simplifying to \(2B = A + C\).

Understanding this property helps us explore the intricate relationship between the angles and the nature of the triangle they form.

Complex Numbers

Complex numbers are a vital concept involving quantities expressed in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit representing \(\sqrt{-1}\). They expand our ability to perform calculations beyond the limitations of real numbers alone. Specifically, complex exponentials such as \(e^{iA}\) utilize Euler's formula, \(e^{ix} = \cos(x) + i\sin(x)\). This allows:

• Representation of a point in the complex plane, where angles correspond to rotations around the unit circle.
• Ease in manipulation of trigonometric expressions, especially when considering properties like A.P.

In the context of triangles, when angles are represented in this way, their alignment in progression offers insights into the geometric properties and structure of the triangle.

Equilateral Triangle

An equilateral triangle is a special type of triangle where all three interior angles are equal, each measuring 60 degrees. This type of triangle also has equal side lengths. In the case of angles forming an arithmetic progression, if calculations reveal equal angles, it confirms the triangle as equilateral. This aligns with these properties:

• Symmetry and equal distribution of angles and sides.
• Every angle is precisely \(60^\circ\), leading to a total of \(3 \times 60^\circ = 180^\circ\), which aligns perfectly with the angle sum property of a triangle.

This results from the condition \(A = B = C\), as demonstrated in the original solution, and demonstrates the inherent balance and uniformity within equilateral triangles.

Triangle Angle Sum

The fundamental property of any triangle is that the sum of its interior angles is always 180 degrees or \(\pi\) radians. This principle is crucial when considering the implications of angles being part of any progression. For a triangle:

• The equation \(A + B + C = \pi\) reflects this geometric truth.
• Any configuration of angles must satisfy this property to form a legitimate triangle.

When exploring angles in arithmetic progression, the sum condition guides how the angles can relate to one another. Understanding this sum is essential to solving any problems related to the type or classification of the triangle, as it sets the groundwork for any logical deductions and solution verification through calculations.