Problem 222 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \left\\{\begin{array}{l} \tan x<1 \\ \cot x \geq-\frac{1}{\sqrt{3}} \end{array}\right\\} $$

Step-by-Step Solution

Answer

The solutions of the inequalities \(\tan x < 1\) and \(\cot x \geq -1/\sqrt{3}\) are \(x \in [\pi/4, \pi/3] \cup [2\pi/3, 5\pi/4]\).

Step 1: Solve the first inequality \(\tan x < 1\)

To solve the inequality \(\tan x < 1\), we look at the unit circle and recognize that the tangent function has values less than 1 in the intervals \(-\pi/4 < x < \pi/4\) and \(\pi/4 < x < 5\pi/4\).

Step 2: Solve the second inequality \(\cot x \geq -1/\sqrt{3}\)

To solve the second inequality, we first rewrite it in terms of tan, yielding \(\tan x \leq -\sqrt{3}\). We find that tan has values less than or equal to \(-\sqrt{3}\) in the intervals \[-\pi/, -2\pi/3] \cup [2\pi/3, \pi]\).

Step 3: Find the intersections of the solution intervals from steps 1 and 2

Now, we need to find the intersection of the solution intervals from Step 1 and Step 2 because both conditions need to be satisfied simultaneously. The intersection of the intervals \(-\pi/4 < x < \pi/4\) and \[\pi/4, 5\pi/4\] from the solution of the first inequality, and the intervals \[-\pi/, -2\pi/3] \cup [2\pi/3, \pi]\) from the solution of the second inequality is \[\pi/4, \pi/3] \cup [2\pi/3, 5\pi/4\]\).

Key Concepts

Inequalities involving tangent and cotangent

When dealing with trigonometric inequalities that involve tangent and cotangent, we're essentially comparing the values of these functions with specific numbers. Tangent, denoted as \(\tan x\), and cotangent, denoted as \(\cot x\), are periodic functions often analyzed over a range of angles on the unit circle.
When solving inequalities like \(\tan x < 1\) or \(\cot x \geq -\frac{1}{\sqrt{3}}\), we seek the values of \(x\) that satisfy these conditions.

• For \(\tan x < 1\), consider the function's period, \(\pi\), and find when \(\tan x\) has values less than 1. This occurs in intervals related to the unit circle, such as \(-\pi/4 < x < \pi/4\) and \(\pi/4 < x < 5\pi/4\).
• For \(\cot x \geq -\frac{1}{\sqrt{3}}\), convert the inequality to employ tangent, as \(\cot x = 1/\tan x\). Thus, analyzing \(\tan x \leq -\sqrt{3}\) helps to pinpoint the relevant angle intervals.

Understanding these inequalities is essential for solving problems that require you to assess where these conditions hold true on the unit circle and within your specified domain.

Intersection of solution sets

Identifying the intersection of two solution sets involves finding common intervals where both conditions of a compound inequality are simultaneously met. When tasked with inequalities such as \(\tan x < 1\) and \(\cot x \geq -\frac{1}{\sqrt{3}}\), you're ultimately finding \(x\) values that satisfy both conditions.
In the context of this problem:

• Solution intervals for \(\tan x < 1\) are \(-\pi/4 < x < \pi/4\) and \(\pi/4 < x < 5\pi/4\).
• Solution intervals for \(\tan x \leq -\sqrt{3}\) are \([-\pi, -2\pi/3] \cup [2\pi/3, \pi]\).

The intersection of these sets is a critical step. By observing where these intervals overlap, the common solutions \([\pi/4, \pi/3] \cup [2\pi/3, 5\pi/4]\) can be highlighted. This tells us where both inequalities are simultaneously true.

Trigonometric functions and unit circle

Understanding the unit circle is crucial when working with trigonometric functions like tangent and cotangent. The unit circle is a circle of radius 1, centered at the origin of a coordinate plane, and is a key tool for visualizing angles and their corresponding sine and cosine values.
The tangent of an angle is the ratio of the sine to cosine of that angle, while the cotangent is the reciprocal of the tangent. In unit circle terms:

• \(\tan x = \frac{\sin x}{\cos x}\)
• \(\cot x = \frac{\cos x}{\sin x}\)

Since sine and cosine values are thus bounded between -1 and 1, tangent and cotangent can vary greatly, highlighting their periodic and often infinite behavior. These functions reach their limits at odd multiples of π/2 (where the cosine is zero), thus making analysis on the unit circle essential for determining their valid ranges in inequality solutions. Understanding this helps us see where tangents and cotangents meet given conditions on the circle, and supports strategies for solving complex inequalities.