Problem 4 - A Complete Resource Book in Mathematics for JEE Main

Question

\(|\tan \theta+\sec \theta|=|\tan \theta|+|\sec \theta|, 0 \leq \theta \leq 2 \pi\) is possible only if (A) \(\theta \in[0, \pi]-\left\\{\frac{\pi}{2}\right\\}\) (B) \(\theta \in[0, \pi]\) (C) \(\theta \in\left[0, \frac{\pi}{2}\right)\) (D) none of these

Step-by-Step Solution

Answer

Option (A) \(\theta \in [0, \pi] - \{\frac{\pi}{2}\}\) is correct.

Step 1: Understand the Equation

The given equation is \(|\tan \theta + \sec \theta| = |\tan \theta| + |\sec \theta|\). We need to identify the condition under which the equality holds true. This equality suggests that either both \(\tan \theta\) and \(\sec \theta\) have the same sign or one of them is zero. This is because if they had opposite signs, their sum inside the absolute value would be less than the sum of their absolute values.

Step 2: Identify Sign Conditions based on Quadrants

Recall the trigonometric signs in different quadrants: In the first quadrant (\(0 \leq \theta < \frac{\pi}{2}\)), both \(\tan \theta\) and \(\sec \theta\) are positive. In the second quadrant (\(\frac{\pi}{2} < \theta \leq \pi\)), \(\tan \theta\) is negative, and \(\sec \theta\) is negative. These quadrants fulfill the sign condition where \(\tan \theta\) and \(\sec \theta\) have the same sign or one is zero.

Step 3: Check Endpoints and Special Angles

Evaluate special angles: At \(\theta = 0\) and \(\theta = \pi\), \(\tan \theta = 0\), and \(\sec \theta = 1\) or \(-1\) which also satisfies the equation. At \(\theta = \frac{\pi}{2}\), \(\sec \theta\) is undefined, so this angle is excluded.

Step 4: Conclusion based on Analysis

Considering the conditions from Steps 2 and 3, the valid range for \(\theta\) is when both \(\tan \theta\) and \(\sec \theta\) share the same sign, specifically between \(0\) to \(\pi\), excluding \(\theta = \frac{\pi}{2}\). Hence, the correct answer is option (A) \(\theta \in [0, \pi] - \{\frac{\pi}{2}\}\).

Key Concepts

Absolute Value

Absolute value represents the non-negative value of a number, effectively disregarding its sign. This is denoted as \(|x|\), where \(x\) can be any real number. The absolute value of \(x\) is always non-negative since it reflects the number's distance from zero on the number line.
In the context of trigonometric functions like \(\tan\theta\) and \(\sec\theta\), absolute values help in evaluating equations or inequalities where signs could affect the calculations. For instance, \(|\tan \theta + \sec \theta| = |\tan \theta| + |\sec \theta|\) indicates a scenario where \(\tan\theta\) and \(\sec\theta\) have identical signs or one of them is zero.
This concept is crucial since if these functions have different signs, their absolute value sum will exceed their combined values within an absolute value. In practical terms, learning about absolute values helps in solving various mathematical problems involving both algebra and trigonometry.

Quadrants in Trigonometry

Quadrants divide the coordinate plane into four sections, each affecting the signs of trigonometric functions differently. Understanding these is key to solving trigonometric equations.
1. In the **first quadrant** \((0 \leq \theta < \frac{\pi}{2})\), all trigonometric functions, including \(\tan\theta\) and \(\sec\theta\), are positive.
2. The **second quadrant** \((\frac{\pi}{2} < \theta \leq \pi)\), where \(\tan\theta\) is negative while \(\sec\theta\) is positive, results in opposing signs for these functions. However, the equation at hand requires matching signs or zero, so these discrepancies rule out certain solutions.
Analyzing the signs in specific quadrants allows us to determine when given trigonometric equations hold true. Selecting the appropriate quadrant ensures accuracy, especially for functions like tangent and secant which behave distinctively in each section.

Trigonometric Identities

Trigonometric identities are equations that relate the trigonometric functions to one another. They play a crucial role in simplifying complex equations and solving trigonometric expressions.
Basic identities include the Pythagorean identities, such as:

• \( \sin^2 \theta + \cos^2 \theta = 1\)
• \(1 + \tan^2 \theta = \sec^2 \theta\)

These identities can be used to transform and simplify expressions wherever needed. In the discussed equation, understanding these identities aids in analyzing whether \(\tan\theta\) and \(\sec\theta\)'s combination follows the condition of sharing the same sign or having one value as zero.
Additionally, recognizing identities helps during problem-solving by reducing complex trigonometric forms into understandable basics. Utilizing these will enhance not just comprehension but also maneuverability across various mathematical problems.