Problem 9 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve for \(x\) : $$ \tan x>\frac{1}{\sqrt{3}} $$

Step-by-Step Solution

Answer

The solution to the inequality \( \tan x > \frac{1}{\sqrt{3}} \) is \( (2n+1)\pi/6 < x < (2n+1)\pi/2 \) where \( n \) is an integer.

Step 1: Identify the quadrant

The problem requires finding the angles for which \( \tan x > \frac{1}{\sqrt{3}} \). The value of \(\frac{1}{\sqrt{3}}\) corresponds to \( \tan(\pi/6) \) or \( \tan(30^\circ) \). Considering the tangent function is positive in both the first and third quadrants, the solution lies in the regions \(\pi/6 < x < \pi/2\) (first quadrant) and \(\pi + \pi/6 < x < \pi + \pi/2\) (third quadrant).

Step 2: Find a general solution

The tan function has a period of \(\pi\), which means every \(\pi\) the function exhibits the same behavior. The general solution for \(x\) is then \( (2n+1)\pi/6 < x < (2n+1)\pi/2 \), where \( n \) is an integer.

Key Concepts

Tangent Function

The tangent function is one of the six primary trigonometric functions and is often written as \( \tan(x) \). This function is crucial in the study of mathematics, especially in geometry and trigonometry. It represents the ratio of the opposite side to the adjacent side in a right-angled triangle:\[ \tan(x) = \frac{\text{opposite}}{\text{adjacent}} \]The tangent function is periodic, meaning it repeats its values at regular intervals, specifically every \( \pi \). This pattern or cycle plays a significant role when finding solutions to trigonometric equations. Another important characteristic of the \( \tan(x) \) function is its behavior in terms of positivity and negativity:

• In the first quadrant, \( \tan(x) \) is positive.
• In the second quadrant, \( \tan(x) \) is negative.
• In the third quadrant, \( \tan(x) \) becomes positive again.
• In the fourth quadrant, it returns to negative.

Knowing these behaviors is essential when solving trigonometric inequalities, as it allows you to pinpoint the ranges where the tangent function meets specific conditions, like being greater than a given value.

Trigonometric Functions

Trigonometric functions are the backbone of trigonometry and include sine, cosine, tangent, cotangent, secant, and cosecant. These functions are derived from the angles and sides of right-angled triangles and are fundamental in various scientific fields, including engineering, physics, and astronomy.Each function is periodic with distinct intervals:

• Sine and cosine are periodic with a cycle of \(2\pi\).
• The tangent and cotangent functions have a period of \(\pi\).
• Secant and cosecant have periods of \(2\pi\).

Characteristics of trigonometric functions:- **Sine and Cosine:** - Reach maximum and minimum values of 1 and -1, respectively. - Are even (cosine) and odd (sine) functions, influencing their symmetry.- **Tangent and Cotangent:** - Do not have maximum or minimum values, as they can extend to infinity. - Are odd functions, showing symmetrical properties across the origin.Understanding these characteristics helps when working with equations and inequalities involving trigonometric functions, like the inequality \( \tan x > \frac{1}{\sqrt{3}} \), by guiding you in selecting the correct intervals and transformations.

General Solution of Trigonometric Equations

Trigonometric equations often require finding all possible solutions, not just within a single cycle of the trigonometric function involved. This is where the concept of the general solution comes in handy. The general solution for a trigonometric equation, like our example \( \tan x > \frac{1}{\sqrt{3}} \), considers the periodic nature of the trigonometric functions. Because tangent repeats every \( \pi \), solutions for \( x \) can be expressed as an infinite set of intervals. For the given inequality, the general solution is derived by:1. Understanding that \( \tan(x) = \tan(x + k\pi) \) for any integer \( k \).2. Using this property to extend solutions to all cycles, not just the angle they're related to.3. Representing them in terms of integer multiples, like \( (2n+1)\frac{\pi}{6} < x < (2n+1)\frac{\pi}{2} \), where \( n \) is an integer.This formulation broadens the solution set and provides a way to systematically identify all possible solutions over the infinite range of values, reflecting the continuous and repeating nature of trigonometric functions. This systematic approach is essential for comprehensively solving trigonometric inequalities.