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

Question

If \(\tan ^{-1}\left(\frac{1}{3}\right)+\tan ^{-1}\left(\frac{1}{4}\right)+\tan ^{-1}\left(\frac{1}{5}\right)+\tan ^{-1}\left(\frac{1}{n}\right)=\frac{\pi}{4}\) where \(n \in N\), then find \(n\).

Step-by-Step Solution

Answer

The number \(n\) is 5.

Step 1: Apply the formula for the sum of two inverse tangent functions

Use the following formula for the sum of tangent inverse functions: \(\tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left( \frac{a + b}{1 - ab} \right)\). Apply this formula for the first two and the second two inverse tangent functions separately. This results in \(\tan^{-1}\left(\frac{1}{3}\right)+\tan^{-1}\left(\frac{1}{4}\right) = \tan^{-1}\left(\frac{7}{11}\right)\) and \(\tan^{-1}\left(\frac{1}{5}\right)+\tan^{-1}\left(\frac{1}{n}\right) = \tan^{-1}\left(\frac{n+5}{5n-1}\right)\).

Step 2: Apply the formula again

Apply the formula for the sum of tangent inverse functions another time to the resulting two inverse tangent functions. This yields \(\tan^{-1}\left(\frac{7}{11}\right)+\tan^{-1}\left(\frac{n+5}{5n-1}\right) = \tan^{-1}\left(\frac{7n+90}{55-7n}\right)\), which needs to be equal to \(\pi/4\) based on the original equation.

Step 3: Apply the inverse property and solve the equation for n

According to the inverse property of tangent, \(\tan^{-1}(x) = y\) is equivalent to \(\tan(y) = x\). Apply this to \(\tan^{-1}\left(\frac{7n+90}{55-7n}\right) = \(\pi/4\) and get \(\tan(\pi/4) = \frac{7n+90}{55-7n}\). Therefore, \(\frac{7n+90}{55-7n} = 1\) since \(tan(\pi/4) = 1\). Solve this equation for \(n\), which gives \(n=5\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the opposite of regular trigonometric functions. They help us find angles when we know the trigonometric ratios.

Let's break down the basics: these functions include \( \tan^{-1}, \sin^{-1}, \cos^{-1}, \) and others. Here, we focus on \( \tan^{-1} \), known as "arctan," which gives the angle whose tangent is a specific value.

Understanding these inverse functions is crucial. Why? They allow us to solve equations involving tangents and angles.

• For instance, if \( \tan(\theta) = x\), then \( \theta = \tan^{-1}(x)\).
• This inverse function returns an angle, usually interpreted to be within the range \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).

Sum of Tangent Inverses

The sum of tangent inverses might seem tricky at first, but there’s a neat formula that makes it simple.

When you want to sum two tangent inverse values, use this formula:
\[ \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1}\left( \frac{a + b}{1 - ab} \right) \]

It's incredibly helpful! Let's see how it works.

• Start with two values, \( \tan^{-1}(\frac{1}{3})\) and \(\tan^{-1}(\frac{1}{4})\).
• Apply the formula: the sum becomes \( \tan^{-1}\left(\frac{\frac{1}{3} + \frac{1}{4}}{1 - \frac{1}{3} \cdot \frac{1}{4}}\right) \). Once you do the math, you find \( \tan^{-1}\left(\frac{7}{11}\right)\).

Another example: sum \( \tan^{-1}(\frac{1}{5})\) and \( \tan^{-1}(\frac{1}{n})\) which becomes \( \tan^{-1}\left(\frac{n+5}{5n-1}\right)\). Repeat these steps as needed.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the involved variables.

They are useful tools, especially when simplifying or solving trigonometric equations.

• A common identity is \( \tan(\pi/4) = 1\). This fact helps in many problems.

In this particular problem, once you get an expression like \( \tan^{-1}\left(\frac{7n+90}{55-7n}\right) \), you equate it to \( \pi/4 \).

By knowing \( \tan(\pi/4) = 1\), you set the expression equal to 1:
\[ \frac{7n+90}{55-7n} = 1 \]

• Solving this equation step-by-step brings you to the solution, which finds \( n = 5 \).

Recognizing and applying these identities can really speed up your problem-solving process! They provide the bridge between angles and ratios.