Problem 29 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Let \(a, b, c\) be positive real numbers and \(\begin{aligned} \theta=\tan ^{-1}\left(\sqrt{\frac{a(a+b+c)}{b c}}\right)+&+\tan ^{-1}\left(\sqrt{\frac{b(a+b+c)}{a c}}\right) \\ &+\tan ^{-1}\left(\sqrt{\frac{c(a+b+c)}{b a}}\right), \end{aligned}\) then the value of \(\tan \theta\) is (a) 0 (b) 1 (c) \(-1\) (d) None

Step-by-Step Solution

Answer

(a) 0

Step 1: Substitute given functions into tan addition formula

Let's denote the three terms under the inverse tangent as \(X\), \(Y\), and \(Z\), such that \(X = \sqrt{\frac{a(a+b+c)}{b c}}\), \(Y= \sqrt{\frac{b(a+b+c)}{a c}}\), and \(Z = \sqrt{\frac{c(a+b+c)}{b a}}\). Therefore, \(\tan \theta = \tan(\tan^{-1}X + \tan^{-1}Y + \tan^{-1}Z)\)

Step 2: Applying the addition formula

We know that \(\tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B}\) so we can apply this formula to \(\theta\), which gives \(\tan \theta = \tan[\tan^{-1}(X + Y) + \tan^{-1}Z] = \frac{X + Y + Z - XY - XZ - YZ}{1 - XZ - YZ - ZX + XYZ}\).

Step 3: Simplify the expression

From the given functions, it can be seen that \(XY = \frac{(a+b+c)^2}{ac}\), \(YZ = \frac{(a+b+c)^2}{ab}\), \(ZX = \frac{(a+b+c)^2}{bc}\), and \(XYZ = a+b+c\). Substituting these values into the expression from Step 2, we get that \(\tan \theta = \frac{0}{1 - (a+b+c)^2 + (a+b+c)} = 0.\)

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are mathematical functions that reverse the action of the basic trigonometric functions such as sine, cosine, and tangent. When a trigonometric function has an angle as its input and provides a ratio of sides in a right-angled triangle as output, the inverse trigonometric function takes this ratio and returns the measure of the angle.

In the context of the IIT JEE trigonometry problem, we encounter the inverse tangent or arctangent function, denoted as \( \tan^{-1} \). When dealing with inverse trigonometric functions, it's important to remember that they output an angle and have restricted domains to maintain their functionality as true functions, since trigonometric functions are periodic and have repeating values.

A key detail often tested in IIT JEE is the manipulation of these functions within complex expressions. It requires a clear understanding that applying \( \tan^{-1} \) to a variable and then taking the tangent of the result will effectively cancel each out, returning the original variable, provided it's in the valid range for the tangent function.

Tan Addition Formula

The tangent addition formula is an essential trigonometric identity that allows us to express the tangent of a sum of two angles in terms of the tangents of each individual angle. The formula is given as \( \tan(A+B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \).

This formula is extremely useful when we need to simplify complex trigonometric expressions, especially in problems that involve the addition of angles, as seen in the given IIT JEE problem. By recognizing situations where the addition formula can be applied, it becomes possible to break down intricate expressions into simpler, more workable parts.

In the given problem, the addition formula is applied in steps to simplify the expression for \( \tan \theta \), eventually leading to the conclusion. Learning how to maneuver these formulas in calculations is crucial for solving high-level trigonometric problems.

Simplifying Expressions

Simplifying expressions in trigonometry often involves using algebraic manipulation alongside trigonometric identities to reduce a complex expression to a more basic form. It is a multi-step process which requires careful substitution and simplification. In the IIT JEE problem, once the tangent of the sum of angles is expressed through the addition formula, the next step entails substituting the known expressions for \(X, Y, Z\) and their products into the new formula.

This process consolidates the expression from a trigonometric one into a purely algebraic one. By breaking down the components into known values and then piecing them together, the expression becomes simplified, often leading to a straightforward answer. In our example, this methodical approach leads to an answer of zero for \( \tan \theta \). It is a skill that's honed with practice, and mastering it is a significant step towards success in exams like the IIT JEE, where complex expressions are commonplace.