Problem 65 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { If } \frac{\sin (x+y)}{\sin (x-y)}=\frac{a+b}{a-b}, \text { then find the value of } \frac{\tan x}{\tan y} \text { in terms of } a \text { and } b \text { . } $$

Step-by-Step Solution

Answer

The short answer to the problem is: \(\frac{\tan x}{\tan y} = \frac{b}{a}\).

Step 1: Rewrite the given equation

To work with the given equation, we can express the sine functions in terms of \(\sin x\), \(\sin y\), \(\cos x\), and \(\cos y\) using the angle-sum and angle-difference formulae. Therefore, we'll rewrite the equation as follows: \[\frac{\sin x \cos y + \cos x \sin y}{\sin x \cos y - \cos x \sin y} = \frac{a+b}{a-b}\]

Step 2: Simplify the equation

To simplify the equation further, we can multiply both the numerator and denominator on the left side by \(\frac{1}{\sin x \cos y}\) which will isolate the terms for \(\tan x\) and \(\tan y\): \[\frac{(\sin x \cos y + \cos x \sin y) \cdot \frac{1}{\sin x \cos y}}{(\sin x \cos y - \cos x \sin y) \cdot \frac{1}{\sin x \cos y}} = \frac{a+b}{a-b} \] The resulting equation becomes: \[\frac{\tan x + \tan y}{\tan x - \tan y} = \frac{a+b}{a-b}\]

Step 3: Solve for \(\frac{\tan x}{\tan y}\)

Now, we'll cross-multiply to isolate the term \(\frac{\tan x}{\tan y}\): \[(\tan x + \tan y)(a-b) = (\tan x - \tan y)(a+b)\] Expanding the equation, we get: \[(\tan x \cdot a - \tan x \cdot b + \tan y \cdot a - \tan y \cdot b) = (\tan x \cdot a + \tan x \cdot b - \tan y \cdot a - \tan y \cdot b)\] Now, we'll take all the terms involving \(\tan x\) to one side and all the terms involving \(\tan y\) to the other side: \[(\tan x \cdot a - \tan x \cdot b + \tan x \cdot a + \tan x \cdot b) = (\tan y \cdot a - \tan y \cdot b + \tan y \cdot a + \tan y \cdot b)\] Which simplifies to: \[2 \tan x \cdot a = 2 \tan y \cdot b\] Divide both sides by \(2ab\): \[\frac{\tan x}{\tan y} = \frac{b}{a}\] At this point, we have found the value of \(\frac{\tan x}{\tan y}\) in terms of \(a\) and \(b\).

Step 4: Write the final answer

From our calculations in the previous steps, we can now write the final answer: \[\frac{\tan x}{\tan y} = \frac{b}{a}\]

Key Concepts

Angle-Sum and Angle-Difference Formulas

The angle-sum and angle-difference formulas are powerful tools in trigonometry. They allow us to express trigonometric functions of sums or differences of angles in terms of functions of individual angles. This is particularly useful when solving equations involving these functions.

For sine, the formulas are:

• Angle-Sum: \(\sin(x+y) = \sin x \cos y + \cos x \sin y\).
• Angle-Difference: \(\sin(x-y) = \sin x \cos y - \cos x \sin y\).

These formulas break down complex trigonometric expressions into simpler parts, making them easier to manage. In our exercise, these formulas help us rewrite the original equation, leading to manageable expressions for further steps. Using these formulas, we cleverly decompose the given complexity into something workable, setting the stage for solving our problem.

Tangent Function

The tangent function, denoted as \(\tan(\theta)\), is a fundamental trigonometric function. It relates to sine and cosine by the relation: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). This function arises frequently in trigonometric identities and equations, particularly those involving right triangles and angle measures.

In our problem, the tangent function emerges when we simplify and manipulate expressions using the angle-sum and angle-difference formulas.

• Through simplification, the original expression leveraging sine becomes one involving tangent: \(\frac{\tan x + \tan y}{\tan x - \tan y}\ = \frac{a+b}{a-b}\).
• This set up allows us to isolate terms involving \(\tan x\) and \(\tan y\), moving closer to our solution.

By understanding the behavior and properties of the tangent function, we are equipped to cross-multiply and simplify to eventually find the ratio \(\frac{\tan x}{\tan y}\) in terms of \(a\) and \(b\).

Cross-Multiplication Technique

Cross-multiplication is a staple arithmetic technique used to solve equations involving ratios or fractions. When given two fractions set equal to each other, this method allows you to clear the fraction by multiplying across the equal sign. This helps isolate variables for an easier solution process.

In the context of this problem, we use cross-multiplication in step 3.

• We start with \(\frac{\tan x + \tan y}{\tan x - \tan y} = \frac{a+b}{a-b}\).
• Cross-multiplying gives us: \( (\tan x + \tan y)(a-b) = (\tan x - \tan y)(a+b)\).

This approach clears the denominators, allowing us to focus on simplifying terms individually. By redistributing and simplifying, cross-multiplication transforms the equation into one where we can effectively separate and solve for \(\frac{\tan x}{\tan y}\). This powerful technique provides clarity and direction when tackling equations with fractions or proportions.