Problem 78 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(\cot A+\cot B+\cot C=\sqrt{3}\), prove that the triangle is equilateral.

Step-by-Step Solution

Answer

We observe that the only case where the condition, \(\cot A+\cot B+\cot C=\sqrt{3}\), can hold true is when all angles A, B, and C are equal to \(\pi/3\). Thus, the triangle is equilateral as all its angles are equal and each angle is 60 degrees.

Step 1: Express cotangent in terms of tangent

We know that \(\cot(x) = 1/\tan(x)\). Therefore, the problem is now \(1/\tan{A} + 1/\tan{B} + 1/\tan{C} = \sqrt{3}\). Meanwhile, in any triangle, \(\tan{A} + \tan{B} + \tan{C} = \tan{A} \cdot \tan{B} \cdot \tan{C}\). Therefore, the given equation becomes \(1/(\tan{A} \cdot \tan{B} \cdot \tan{C}) = \sqrt{3}\)

Step 2: Calculation and Conclusion

And as \(\tan{A} + \tan{B} + \tan{C} = \tan{A} \cdot \tan{B} \cdot \tan{C}\), we can simplify the equation to \(\sqrt{3} = \tan{A} + \tan{B} + \tan{C}\).The only way this can hold true is if \(\tan{A} = \tan{B} = \tan{C} = \sqrt{3}/3\). As all three angles are equal and their sum is \(\pi\), each angle must be \(\pi/3\), which makes the triangle equilateral.

Key Concepts

Trigonometric Identities

Trigonometric identities are key relationships that exist between the trigonometric functions like sine, cosine, tangent, and cotangent. These identities are invaluable tools for solving equations and proving general properties about angles and triangles. One of the most common identities used is a transformation between tangent and cotangent, where

• \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)
• \( \cot(x) = \frac{1}{\tan(x)} = \frac{\cos(x)}{\sin(x)} \)

This way, \( \cot(x) \) can easily be converted to a reciprocal form of \( \tan(x) \). In trigonometry, using these reciprocal identities helps to simplify complex expressions and solve equations effectively. This simplification was crucial in the provided exercise to express \( \cot A + \cot B + \cot C \) in terms of tangent functions, leading to a more manageable form.

Properties of Equilateral Triangles

An equilateral triangle is a special kind of triangle where all three sides are equal in length, and all three internal angles are equal, each measuring \(60^\circ\) or \( \frac{\pi}{3} \) radians. The properties of equilateral triangles include:

• All sides have the same length.
• All angles are equal and add up to \(180^\circ\) or \( \pi \) radians.
• The height creates two 30-60-90 right triangles within the equilateral triangle.

In the given problem, proving that the triangle is equilateral involves showing that the angles are all \( \frac{\pi}{3} \). This conclusion comes from solving the equation where \( \tan A + \tan B + \tan C = \sqrt{3} \), noting that when triangles have equal sides, their opposite angles must also be equal, leading to the requirement that all angles are \(60^\circ\).

Cotangent and Tangent Relationships

The relationship between cotangent and tangent is under a fundamental identity where \( \cot(x) = \frac{1}{\tan(x)} \). This identity is crucial when you need to convert or manipulate expressions that involve tangent and cotangent. In the provided exercise, making use of this relationship formed the basis for simplifying and proving the equilateral nature of the triangle:

• Starting with \( \cot A + \cot B + \cot C = \sqrt{3} \)
• Transform it using \( \cot(x) = \frac{1}{\tan(x)} \)
• The result was \( \tan A + \tan B + \tan C = \tan A \cdot \tan B \cdot \tan C \)
• This leads to \( \tan A = \tan B = \tan C = \frac{\sqrt{3}}{3} \)

These transformations help determine that \( \angle A = \angle B = \angle C = \frac{\pi}{3} \), confirming the triangle's equilateral properties. Understanding and applying these relationships makes solving such trigonometric problems much easier.