Problem 271 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \tan ^{-1} x>-\frac{\pi}{3} $$

Step-by-Step Solution

Answer

The solution set of the given inequality is \(x > -\sqrt{3}\).

Step 1: Understand the arctan function

In order to solve this inequality, it's important to understand the arctan function. Arctan, or \(\tan^{-1} x\), is a function that returns the angle whose tangent is x. The result is in the interval \(-\pi/2\) to \(\pi/2\).

Step 2: Apply the arctan property

Since we need to solve for x in \(\tan ^{-1} x > -\frac{\pi}{3}\), we can apply the property of arctan, that is, \(\tan(\tan ^{-1} x) = x\). Applying this property on both sides of the inequality, we get \(x > \tan(-\frac{\pi}{3})\) since tan is an increasing function.

Step 3: Evaluate the right side

Evaluate the right side of the equation, \(x > \tan(-\frac{\pi}{3})\) to get the value of x. We know that \(\tan(-\frac{\pi}{3}) =-\sqrt{3}\). Thus, the inequality becomes \( x > -\sqrt{3}\). That is, x is greater than \(-\sqrt{3}\).

Step 4: Write the solution

The solution to the inequality is \(x > -\sqrt{3}\). This represents all real numbers greater than \(-\sqrt{3}\).

Key Concepts

Arctan Function

Understanding the Arctan Function is key when solving certain types of trigonometric inequalities. The arctan function, also known as the inverse tangent function, is written as \( \tan^{-1} x \). It is the inverse of the tangent function. Essentially, it tells you the angle whose tangent is a given value \( x \). With arctan, the resulting angle is always between \( -\frac{\pi}{2} \) and \( \frac{\pi}{2} \).

• This means that for any real number input \( x \), the function will output an angle within that range.
• For example, if \( \tan^{-1} (x) = \theta \), then \( \tan(\theta) = x \).
• One key feature of arctan, like all inverse trigonometric functions, is its bijection property on its restricted range.

Knowing this, we can proceed to solve inequalities involving \( \tan^{-1} x \) with a clear understanding of its constraints and outputs.

Inequality Solving

Inequality Solving skills are crucial when dealing with arctan functions, especially in the context of trigonometric inequalities. An inequality is a mathematical statement that shows the relationship between two expressions that are not equal, using inequality signs like \( >, <, \geq, \leq \).
When faced with an inequality like \( \tan^{-1} x > -\frac{\pi}{3} \), the first step is understanding the function's behavior and range. Since the arctan function is increasing, if \( \tan^{-1} x > -\frac{\pi}{3} \), then the tangent of both sides keeps the inequality sign intact.

• "Increasing" means if \( a > b \), then \( \tan(a) > \tan(b) \).
• By applying tangent to both sides: \( \tan(\tan^{-1}(x)) > \tan(-\frac{\pi}{3}) \).

This simplifies to evaluating \( x > \tan(-\frac{\pi}{3}) \). Through solving such inequalities, one often finds solutions that describe entire ranges of values for \( x \)."

Tangent Function Properties

The Tangent Function Properties are fundamental in understanding how to solve inequalities like \( \tan^{-1} x > -\frac{\pi}{3} \). The tangent function, denoted as \( \tan \), relates the angle of a right-angled triangle to the ratio of the opposite side over the adjacent side. It is periodic with a period of \( \pi \), and unlike sine and cosine, the tangent function can take any real value.

• One critical property of tangent is that it is an increasing function. This is why, when solving \( x > \tan(-\frac{\pi}{3}) \), the inequality remains \( x > -\sqrt{3} \).
• The tangent of negative angles yields negative values, which is why \( \tan(-\frac{\pi}{3}) = -\sqrt{3} \).
• Understanding these properties helps quickly determine that the solution \( x > -\sqrt{3} \) covers all real numbers greater than \( -\sqrt{3} \).

The increasing nature of the tangent function ensures that for \( x > y \), also \( \tan(x) > \tan(y) \), aiding us in maintaining the inequality direction while solving.