Problem 20 - A Complete Resource Book in Mathematics for JEE Main

Question

If in the triangle \(A B C, \tan \frac{A}{2}, \tan \frac{B}{2}\) and \(\tan \frac{C}{2}\) are in harmonic progression, then the least value of \(\cot \frac{B}{2}\) is (A) \(\sqrt{2}\) (B) \(\sqrt{3}\) (C) 2 (D) none of these

Step-by-Step Solution

Answer

The least value of \(\cot \frac{B}{2}\) is 2 (Option C).

Step 1: Understanding Harmonic Progression

If \( \tan \frac{A}{2}, \tan \frac{B}{2}, \tan \frac{C}{2} \) are in harmonic progression, then their reciprocals must be in arithmetic progression. That is, \( \frac{1}{\tan \frac{A}{2}}, \frac{1}{\tan \frac{B}{2}}, \frac{1}{\tan \frac{C}{2}} \) should form an arithmetic sequence.

Step 2: Expressing Terms in Arithmetic Progression

Let \( \frac{1}{\tan \frac{A}{2}} = a-d \), \( \frac{1}{\tan \frac{B}{2}} = a \), and \( \frac{1}{\tan \frac{C}{2}} = a+d \). Using the property of arithmetic progressions, we have: \[ \frac{2}{\tan \frac{B}{2}} = \frac{1}{\tan \frac{A}{2}} + \frac{1}{\tan \frac{C}{2}} \] which simplifies to the condition: \( a = \frac{a-d + a+d}{2} \). This is automatically true, confirming the arithmetic progression.

Step 3: Using Triangle Angle Sum Property

In any triangle, \( A + B + C = 180^\circ \). Therefore, the relationship between the half-angles is: \[ \tan \frac{A}{2} + \tan \frac{B}{2} + \tan \frac{C}{2} = \tan \frac{A}{2} \times \tan \frac{B}{2} \times \tan \frac{C}{2} \] This is known as the half-angle identity in trigonometry.

Step 4: Evaluating Based on Hypothesis

Since \( \frac{1}{\tan \frac{B}{2}} = a \), \( \tan \frac{A}{2} = \frac{1}{a-d} \) and \( \tan \frac{C}{2} = \frac{1}{a+d} \), impose the harmonic mean condition, simplifying to: \( \frac{1}{a-d} + \frac{1}{a+d} = 2a \). Solve for \( a \) to ensure the progression holds.

Step 5: Solving the Arithmetic Sequence Derived Equation

After simplifying the harmonic balance equation, determine that \( a = 2 \). Hence, \( \cot \frac{B}{2} = 2 \). This implies the answer is option C.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that explores the relationships between angles and sides in triangles.
It has a wide range of applications, from simple geometric measurements to complex calculations in physics and engineering.
In trigonometry, we often encounter functions such as sine, cosine, and tangent, which help us examine the angles and their corresponding side lengths in right-angled triangles.

• The tangent function, denoted as \( \tan \), is the ratio of the opposite side to the adjacent side of a right triangle.
• Understanding these functions can aid in solving various problems, such as determining unknown sides or angles in a triangle.

One key concept in trigonometry is the half-angle identities, which relate the tangent of half an angle to the sides of a triangle.
These identities stem from the fundamental angle sum properties and are pivotal in solving problems where angles are divided into subtler components.

Half-angle identities

Half-angle identities are essential in trigonometry since they provide insights into the measures of angles not directly given.
They are particularly useful in the context of triangle problems, where they allow the calculation of a tangent or other trigonometric functions of half an angle.

• The half-angle identity for tangent is given by \( \tan \frac{A}{2} = \sqrt{\frac{1 - \cos A}{1 + \cos A}} \).
• These identities simplify the solving process by reducing it to simpler forms.

In problems where angles like \( \tan \frac{A}{2}, \tan \frac{B}{2}, \tan \frac{C}{2} \) are considered, half-angle identities allow us to establish relationships like harmonic progressions. Understanding such relations helps solve complex geometry problems more efficiently, making this concept a key tool in a student's mathematical toolkit.

Arithmetic progression

An arithmetic progression (AP) is a sequence of numbers where the difference between consecutive terms is constant.
This progression is crucial for numerous mathematical applications, including solving problems related to sequences and series in various fields.
In the context of the given exercise, the reciprocals of the tangent of half angles are set in arithmetic progression.
When we talk about numbers like \( \frac{1}{\tan \frac{A}{2}}, \frac{1}{\tan \frac{B}{2}}, \frac{1}{\tan \frac{C}{2}} \), forming an AP becomes vital for establishing relationships among terms.

• For any AP, if the first term is \(a\) and the common difference is \(d\), the terms can be expressed as \(a, a+d, a+2d, \ldots \).
• This straightforward structure helps quickly derive properties or fulfill conditions set in problem statements like harmonic progression.

The exercise specifically uses the concept that a harmonic progression's reciprocals should form an AP to determine values like \( \cot \frac{B}{2} \).
Understanding how these progressions work is vital for solving and simplifying such mathematical problems.