Problem 14 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

\(\tan ^{-1}\left(\frac{1}{2}\right)+\tan ^{-1}\left(\frac{1}{3}\right)\) is equal to (a) \(\frac{\pi}{4}\) (b) \(\frac{\pi}{2}\) (c) \(\frac{\pi}{3}\) (d) None of these

Step-by-Step Solution

Answer

The correct answer is (a) \( \frac{\pi}{4} \).

Step 1: Apply the formula

The formula for the sum of two inverse tangent functions is given by: \(\tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1} \left( \frac{a+b}{1-ab} \right)\). By applying this formula to the problem at hand, where \(a = \frac{1}{2}\), and \(b = \frac{1}{3}\), we get: \(\tan^{-1}\left(\frac{1}{2}\right)+\tan^{-1}\left(\frac{1}{3}\right) = \tan^{-1} \left( \frac{\frac{1}{2}+\frac{1}{3}}{1-\frac{1}{2}*\frac{1}{3}} \right)\).

Step 2: Solve the fractions

Now, simplifying the expression inside the inverse tangent yields \(\tan^{-1} \left( \frac{ \frac{5}{6}}{1 - \frac{1}{6}} \right) = \tan^{-1} (1)\).

Step 3: Recognize the angle

From trigonometry, it's known that \(\tan^{-1}(1) = \frac{\pi}{4}\). Hence, the sum \(\tan^{-1}\left(\frac{1}{2}\right)+\tan^{-1}\left(\frac{1}{3}\right) = \frac{\pi}{4}\).

Key Concepts

Trigonometric Identities

Trigonometric identities are formulas that express relationships between the trigonometric functions such as sine, cosine, tangent, etc. These identities are incredibly useful for solving problems in trigonometry.
One key identity used in the exercise above is the sum of two inverse tangent functions. This is a less common identity but very helpful: \( \tan^{-1}(a) + \tan^{-1}(b) = \tan^{-1} \left( \frac{a+b}{1-ab} \right) \).
This equation helps us to combine two inverse tangents into one, simplifying the process of finding the angle these tangents are associated with. For example, in the exercise, by substituting \(a = \frac{1}{2}\) and \(b = \frac{1}{3}\), it allowed us to reduce the expression to just one inverse tangent, thus making the problem easier to solve. Understanding these identities can simplify many problems that initially seem complex.

Tangent Function

The tangent function, often abbreviated as just 'tan', is a fundamental concept in trigonometry. Unlike sine and cosine, which relate the length of the sides to the angle, the tangent function relates the ratio of the opposite side to the adjacent side in a right-angled triangle.
This function is periodic, with a period of \(\pi\). The tangent function can take any real number as an input and provide outputs ranging from negative to positive infinity.
In our exercise, the inverse tangent, denoted as \(\tan^{-1}(x)\), was used. This function finds the angle whose tangent is \(x\). It is by using \(\tan^{-1}(1)\), we were able to determine that the angle in question was \(\frac{\pi}{4}\).

• Range of \(\tan^{-1}\): The range is between \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\).
• Common Values: Knowing values such as \(\tan^{-1}(1) = \frac{\pi}{4}\) is useful in problem-solving.

Problem Solving in Trigonometry

Solving trigonometric problems can initially appear daunting, but understanding fundamental concepts like the ones used here can simplify the process.
When solving such problems, follow these steps:

• Identify what is given: In our case, the problem provided two inverse tangent values.
• Apply relevant identities: Use trigonometric identities to simplify or combine these functions as demonstrated with the sum identity.
• Simplify and evaluate: Breaking down fractions and simplifying expressions, as we converted to \(\tan^{-1}(1)\), simplifies the process.
• Use known values: Recognizing common results like \(\tan^{-1}(1) = \frac{\pi}{4}\) helps in quick calculations.

Practice is critical to mastering these techniques and gaining confidence in applying these identities and functions.