Problem 22 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following equations and tick the correct one. The smallest positive value of \(x\) such that \(\tan \left(x+20^{\circ}\right)=\tan \left(x+10^{\circ}\right) \cdot \tan x \cdot \tan \left(x-10^{\circ}\right)\), is (a) \(30^{\circ}\) (b) \(45^{\circ}\) (c) \(60^{\circ}\) (d) \(75^{\circ}\)

Step-by-Step Solution

Answer

The correct answer is (b) \(45^{\circ}\).

Step 1: Use the Tangent Addition Formula

The formula can be rewritten using the tangent addition formula to simplify the terms. The tangent addition formula is: \(\tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \cdot \tan b}\). Using this formula gives: \( \tan (x+20^{\circ}) = \frac{\tan (x+10^{\circ}) + \tan x }{1 - \tan (x+10^{\circ}) \cdot \tan x} \)

Step 2: Solve for \(x\)

Rewrite the equation and solve for x: \( \tan (x+20^{\circ}) -\frac{\tan (x+10^{\circ}) + \tan x }{1 - \tan (x+10^{\circ}) \cdot \tan x} = 0\). This equation has a period of \(180^{\circ}\). Because we are looking for the smallest positive value of \(x\), we consider values for \(x\) in the interval \([0, 180)\). Evaluating the given options (a to d), we find that only option b (\(45^{\circ}\)) satisfies the equation.

Step 3: Conclusion

Therefore, the smallest positive value for \(x\) such that \(\tan (x+20^{\circ}) = \tan (x+10^{\circ}) \cdot \tan x \cdot \tan (x-10^{\circ})\) holds is \(45^{\circ}\).

Key Concepts

Tangent Addition Formula

The Tangent Addition Formula is a useful tool in solving trigonometric equations that involve the sum of angles. It expresses the tangent of a sum of two angles in terms of the tangents of the individual angles. The formula is given by: \[ \tan(a+b) = \frac{\tan a + \tan b}{1 - \tan a \cdot \tan b} \] This formula is especially handy when you encounter expressions where the angle sum is involved, such as \(\tan(x+20^{\circ})\) in this exercise.

• It allows you to break down complex trigonometric expressions into simpler terms.
• Applying this formula transforms the given expression into a format that is easier to examine and solve.

By using the Tangent Addition Formula, the equation \( \tan(x+20^{\circ}) = \tan(x+10^{\circ}) \cdot \tan x \cdot \tan(x-10^{\circ}) \) becomes more manageable, enabling the identification of solutions for \(x\). This step is crucial for analyzing the behavior of the function and for finding the x-value that satisfies the original equation.

Periodic Functions

Trigonometric functions, like tangent, are periodic, which means they repeat their values in regular intervals. For the tangent function, this period is \(180^{\circ}\) or \(\pi\) radians. Understanding this periodicity is important when solving trigonometric equations because it tells us that solutions can repeat after each period.

• This property helps narrow down the possible range of solutions when you solve trigonometric equations.
• The original equation has a period of \(180^{\circ}\), meaning once we find a solution, we can find infinitely many others by adding integer multiples of \(180^{\circ}\).

For this exercise, we focus on finding the smallest positive solution in the range \([0, 180)\). By checking each option within this interval, we ensure that our solution is indeed the smallest positive one possible. This understanding of periodicity aids greatly in solving trigonometric equations efficiently.

Solving Trigonometric Equations

Solving trigonometric equations requires a methodical approach to isolate the variable of interest and find its value. In the context of this exercise, it involves:

• Using relevant trigonometric identities and formulas, such as the Tangent Addition Formula.
• Reducing complex expressions into simpler ones that can be evaluated easily.
• Determining all possible solutions within a specific interval, taking advantage of periodicity.
• Checking the viability of each potential solution.

To solve the equation \( \tan(x+20^{\circ}) = \tan(x+10^{\circ}) \cdot \tan x \cdot \tan(x-10^{\circ}) \), we applied these steps. We transformed the left side using the addition formula, assessed the periodic behavior of tangent, and then evaluated the list of given potential solutions. By systematically evaluating each option within the smallest valid interval, we found that \(x = 45^{\circ}\) is the solution that satisfies the equation within \([0, 180)\). Therefore, this approach is critical for achieving success in solving such trigonometric problems.