Problem 36 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

If \(x<0\), then \(\tan ^{-1}\left(\frac{1}{x}\right)\) is (a) \(\cot ^{-1}(x)\) (b) \(-\cot ^{-1}(x)\) (c) \(-\pi+\cot ^{-1}(x)\) (d) None

Step-by-Step Solution

Answer

The correct answer is (b) \(-\cot^{-1}(x)\).

Step 1: Understanding the inverse trigonometric relations

Remember the relationship between \(\tan^{-1}(x)\) and \(\cot^{-1}(x)\): they are simple inverses of one another. Thus, \(\tan^{-1}(x) = \frac{\pi}{2} - \cot^{-1}(x)\) and similarly, \(\cot^{-1}(x) = \frac{\pi}{2} - \tan^{-1}(x)\).

Step 2: Solve for given equation

For this exercise, the task is to identify what \(\tan^{-1}(1/x)\) equals to when \(x<0\). We know that \(\tan^{-1}(1/x) = \cot^{-1}(x)\). Since \(x<0\), \(\cot^{-1}(x)\) will be negative, hence \(\tan^{-1}(1/x) = -\cot^{-1}(-x)\). Therefore, the correct solution is \(-\cot^{-1}(-x)\) which corresponds to option (b) when \(x < 0\).

Key Concepts

Cotangent Inverse

The inverse cotangent function, denoted as \(\cot^{-1}(x)\), is one of the inverse trigonometric functions. It reverses the action of the cotangent function, meaning that if \(\cot(\theta)=x\), then \(\theta=\cot^{-1}(x)\). For the domain of real numbers, the range of \(\cot^{-1}(x)\) is typically defined from \(0\) to \(\pi\), excluding the point \(\frac{\pi}{2}\) where cotangent is undefined because the tangent of \(\frac{\pi}{2}\) is infinity.

Due to its periodic nature, the cotangent function can have many angles that produce the same cotangent value. Therefore, its inverse is carefully defined to provide a unique output for every input value. In applications, \(\cot^{-1}(x)\) is often seen in integration and in solving trigonometric equations. Understanding how it interacts with signs of 'x' is crucial for correctly interpreting its values, which is often tested in exams like IIT JEE.

Tangent Inverse

The inverse tangent function, often written as \(\tan^{-1}(x)\) or \(\arctan(x)\), is the inverse operation of the tangent function. When you apply the tangent function to an angle, you get the ratio of the opposite side to the adjacent side in a right-angled triangle. Conversely, when you apply \(\tan^{-1}(x)\), it tells you the angle whose tangent is 'x'.

The range of \(\tan^{-1}(x)\) is from \(\frac{-\pi}{2}\) to \(\frac{\pi}{2}\). It's a continuous function, which means there aren't any breaks or jumps in its values. In the context of the IIT JEE trigonometry syllabus, the inverse tangent function is crucial for solving a variety of problems, especially those involving integration or finding angles from given tangent values.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. They play a fundamental role in simplifying trigonometric expressions and solving equations. Some of the most essential trigonometric identities include:

• Pythagorean identities, such as \(\sin^2(x) + \cos^2(x) = 1\)
• Angle sum and difference identities, for example, \(\sin(x \pm y) = \sin(x)\cos(y) \pm \cos(x)\sin(y)\)
• Double angle identities like \(\sin(2x) = 2\sin(x)\cos(x)\)
• Reciprocal identities, such as \(\tan(x) = \frac{\sin(x)}{\cos(x)}\) and \(\cot(x) = \frac{\cos(x)}{\sin(x)}\)

Mastering these identities is vital for solving complex trigonometric problems and is a must-have skill for students preparing for competitive exams like IIT JEE, where trigonometry is a significant component.

IIT JEE Trigonometry

Trigonometry is a substantial part of the mathematics curriculum for IIT JEE, a prestigious entrance examination for engineering institutes in India. The subject tests a student's ability to understand and manipulate trigonometric functions, graphs, identities, and equations.

The IIT JEE syllabus covers a broad range of topics, including:

• Measurement of angles and conversion between different units of angle measurement
• Graphs and properties of trigonometric functions
• Inverse trigonometric functions and their properties
• Solving trigonometric equations
• Utilization of trigonometric identities to simplify and solve problems

Students are expected to have a deep understanding of trigonometric concepts and demonstrate problem-solving skills through various challenging questions, such as those involving inverse trigonometric functions, which require a good grasp of their behaviors and properties.