Problem 3 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following trigonometric equations: \(\frac{\tan x}{\tan 2 x}+\frac{\tan 2 x}{\tan x}+2=0\)

Step-by-Step Solution

Answer

The solutions for the equation are \(x = \tan^{-1}(\sqrt{5/3})\) and \(x=\tan^{-1}(-\sqrt{5/3})\).

Step 1: Apply the identity of tan 2x

We can start by replacing \(\tan 2x\) in the equation with its identity \(2\tan x / (1 - \tan^2 x)\). Our equation becomes \((\tan x / (2\tan x / (1 - \tan^2 x))) + (2\tan x / \tan x) / \tan x +2=0\). We simplify it by multiplying both the numerator and denominator by \((1 - \tan^2 x)\) in the first term, which gives us \((1 - \tan^2 x + 2(1 - \tan^2 x) + 2=0\)

Step 2: Simplify the equation

By combining like terms, we get \(1 - \tan^2 x + 2 - 2\tan^2 x + 2 = 0\). When we simplify further, we get \(-3\tan^2 x + 5 = 0\).

Step 3: Solve for tan x

To find the solution for \(\tan x\), we can simply rearrange the equation to get \(\tan^2 x = 5/3\). Taking square root on both sides, we get \(\tan x = \sqrt{5/3}\). It could also be negative, which indicates that there are 2 possible solutions for the equation.

Key Concepts

Understanding tan 2x

The function \( \tan 2x \) represents a well-known double angle trigonometric identity. This identity is crucial when solving this equation because it offers a way to express \( \tan 2x \) using \( \tan x \). The identity is given by: \[ \tan 2x = \frac{2\tan x}{1 - \tan^2 x} \] This identity helps us substitute \( \tan 2x \) in terms of \( \tan x \), making it easier to manipulate and work within equations.

• \( \tan 2x \) is not just \( 2 \times \tan x \), it's more complex due to its derivation from the sine and cosine ratio definition.
• Using this double angle formula, we express, simplify, and solve equations that include \( \tan 2x \).

This function transformation science can seem tricky at first, but enhancing your understanding of these identities allows for easier equation solving.

The Role of Trigonometric Identities

Trigonometric identities are tools that allow us to transform and solve trigonometric equations more easily. They are essential when trying to convert or simplify complex trigonometric equations like the one provided here. In our case, the identity of \( \tan 2x \) was applied to rewrite the equation initially given. A trigonometric identity relates one trigonometric function to others, allowing the equation to be simplified or restructured. Besides the double angle identity discussed, there are others like Pythagorean identities, which can also be useful:

• Pythagorean Identity: \( \sin^2 x + \cos^2 x = 1 \).
• Sum to Product Identities.

Understanding these identities allows us to simplify seemingly complex equations into more manageable forms, which, in turn, makes solving them much more feasible.

Finding Solutions to Trigonometric Equations

Solving a trigonometric equation involves finding all possible values of the variable that satisfies the given equation. The initial problem led us to an equation in terms of \( \tan^2 x \). Once the equation \( -3\tan^2 x + 5 = 0 \) was established, it was simplified to find \( \tan x \):

• Rearrange terms: \( \tan^2 x = \frac{5}{3} \)
• Take the square root: \( \tan x = \sqrt{\frac{5}{3}} \) or \( \tan x = -\sqrt{\frac{5}{3}} \)

This step shows that \( \tan x \) can have two values based on the positive and negative roots. This is because squaring a number loses the sign, hence both positive and negative roots must be considered. Examining within typical trigonometric function cycles or specified ranges ensures comprehensiveness of solutions.