Problem 18 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(A>0, B>0\) and \(A+B=\frac{\pi}{3}\), find the maximum value of \(\tan A \tan B\).

Step-by-Step Solution

Answer

There is no clear maximum value for \(\tan A \tan B\) because the range of \(\tan x\) spans from \(-\infty\) to \(\infty\) and the function oscillates indefinitely.

Step 1: Conversion to another Trigonometric Function

To simplify the expression, convert \(\tan A \tan B\) to another trigonometric function using the identity \(\tan x \tan y = \frac{\sin(2x) -\sin(2y)}{\cos(2x) + \cos(2y)}\). Apply this to get this: \(\tan A \tan B = \frac{\sin(2A) -\sin(2B)}{\cos(2A) + \cos(2B)}\).

Step 2: Simplifying the Expression

Recognize the numerator and denominator as sum and difference formulas, this can be expressed as: \(\tan A \tan B = \frac{\sin[(2A + 2B) - (2A - 2B)]}{\cos[(2A + 2B) - (2A - 2B)]}\). We know that \(A + B = \frac{\pi}{3}\), so \(2A + 2B = \frac{2\pi}{3}\). Substituting this, we get: \(\tan A \tan B = \frac{\sin[2\frac{\pi}{3}]}{\cos[2\frac{\pi}{3}]} = \tan(2\frac{\pi}{3})\).

Step 3: Evaluating the Maximum Value

The range of \(\tan x\) is from \(-\infty\) to \(\infty\). Although the function oscillates, a clear maximum within the given bounds is not present. Therefore, it cannot be determined by conventional means.

Key Concepts

Sum and Difference Formulas

The sum and difference formulas in trigonometry are essential tools that allow us to simplify expressions involving trigonometric functions of multiple angles. These formulas can help express the sine, cosine, and tangent of sums or differences of angles in terms of the functions of individual angles.

For example, the sum formula for sine states: \[\sin(A + B) = \sin A \cos B + \cos A \sin B.\] Similarly, the difference formula for cosine is given by: \[\cos(A - B) = \cos A \cos B + \sin A \sin B.\]These identities are particularly useful when you have an equation involving angles like in the original exercise where \(A + B = \frac{\pi}{3}\). This simplifies the task of substituting numeric values into trigonometric functions to find unknown values or maximums, as was necessary to express \(\tan A \tan B\) in another trigonometric form using the identity. This indirect method aids in handling more complex expressions efficiently.

Understanding these formulas not only allows for algebraic manipulations but also enhances the understanding of periodic behaviors of trigonometric functions, which is invaluable in calculus and analytic trigonometry.

Maximum Value Problems

Maximum value problems are an interesting aspect of mathematical analysis where the aim is to find the highest point or value that a function can achieve. In the case of trigonometric functions, we often deal with cyclic and periodic behaviors, which can pose unique challenges in determining maximum values.

In trigonometry, especially with tangent functions, there's usually an infinite range, which makes pinpointing a maximum tricky. This exercise asks for the maximum value of \( \tan A \tan B \), where \( A+B = \frac{\pi}{3} \). The problem involves understanding how tangent values behave on intervals and how to apply maximum and optimization strategies within such constraints.

To approach these types of problems, it's helpful to break down the function into simpler components, using identities or expressions like those mentioned in the solution. While tangent itself doesn't have a bounded maximum, by linking it to known angles and identities, we can sometimes deduce behaviors, even if a conventional maximum isn’t attainable. Such deductions are crucial in fields like engineering and physics where practical solutions are necessary, despite mathematical limitations.

Tangent Function Properties

The tangent function, \( \tan x \), is one of the primary trigonometric functions and has some unique properties that distinguish it from others, such as sine and cosine. Understanding these properties is vital when working with problems involving tangent, as they influence how you interpret and manipulate the function.

Key properties include:

• Tangent is periodic, with a period of \( \pi \), meaning \( \tan(x + \pi) = \tan x \).
• The function is undefined at angles where the cosine is zero, such as \( \frac{\pi}{2} + k\pi \) for any integer \( k \), leading to vertical asymptotes.
• Tangent exhibits odd symmetry, so \( \tan(-x) = -\tan(x) \), which is useful in simplifying expressions.

For the given problem, understanding the behavior of tangent around the angle \( \frac{\pi}{3} \), and its relation to other angle measures, can clarify how \( \tan A \tan B = \tan(\frac{2\pi}{3}) \) was derived after substitution. Knowing that \( \tan \) values oscillate between approach towards infinity and negative infinity within any given period assists in comprehending why a maximum value isn’t straightforwardly attainable in the same sense as with sine or cosine.