Problem 5 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Which one of the following is correct? (a) \(\tan 1>\tan ^{-1} 1\) (b) \(\tan 1<\tan ^{-1} 1\) (c) \(\tan 1=\tan ^{-1} 1\) (d) None.

Step-by-Step Solution

Answer

The correct answer is (a): \(\tan(1) > \tan^{-1}(1)\)

Step 1: Calculate tan(1)

As the tangent function is continuous and differentiable in its domain, you can calculate its value at 1, leaving the result in radian form: tan(1) \approx 1.557...

Step 2: Calculate inverse tan(1)

Inverse tangent function, also called arctangent, is the inverse function of the tangent. Arctangent of 1, often written as \(\tan^{-1}(1)\) or atan(1), is equal to \(\frac{\pi}{4}\) radians, or approximately 0.785...

Step 3: Compare the values

Having calculated both values, we find that \(\tan(1) > \tan^{-1}(1)\). This confirms statement (a) as true.

Key Concepts

Tangent Function

The tangent function is one of the fundamental trigonometric functions, often written as \( \tan(\theta) \). It relates the angle \( \theta \) to the ratio of the opposite side to the adjacent side in a right-angled triangle.
There are some important properties of the tangent function:

• It has a period of \( \pi \), meaning that it repeats every \( \pi \) radians.
• The function is undefined at odd multiples of \( \pi/2 \) since the cosine of these angles is zero.
• It takes on both positive and negative values, depending on the angle.
• The tangent function is continuous and differentiable in its domain.

For small angles, the tangent function can be approximated by the angle itself when considered in radians. This approximation is useful in engineering and physics, where angles are often small. In our exercise, when calculated at 1 radian, \( \tan(1) \approx 1.557 \), showing it is larger than \( \tan^{-1}(1) \).

Arctangent Function

The arctangent function, also known as \( \tan^{-1}(x) \) or \( \text{atan}(x) \), is the inverse of the tangent function. This function returns an angle whose tangent is a given number. Its primary purpose is to determine an angle when the tangent of that angle is known.
Key points to understand the arctangent function:

• The range of the arctangent function is \( (-\pi/2, \pi/2) \) since it only returns angles within this interval.
• It is a continuous function, meaning there are no breaks or jumps in its graph.
• The arctangent is useful in inverse trigonometric equations to find angles.

An example of its application can be seen in our exercise, where \( \tan^{-1}(1) = \pi/4 \text{ or approximately 0.785}... \) radians. This demonstrates its role in solving equations involving tangent values.

Trigonometric Inequalities

Trigonometric inequalities are expressions involving trigonometric functions, where one function is greater or less than another. Solving these inequalities often requires understanding the behavior of trigonometric functions over their domains.
When dealing with such inequalities:

• Analyze the values of the trigonometric functions in the specific interval.
• Consider the fundamental period and properties of these functions.
• Approximations may be performed to simplify solutions when the angle is given in radians.

In this context, the inequality \( \tan(1) > \tan^{-1}(1) \) is an example, where both functions are evaluated and compared. The result \( \tan(1) \approx 1.557 \) is greater than \( 0.785 \), showing the inequality holds true. Understanding this process can help tackle other inequalities involving trigonometric functions.