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

Question

Solve for \(x\) : $$ \cos x>\frac{\sqrt{3}}{2} $$

Step-by-Step Solution

Answer

The solution to the inequality \(\cos x > \frac{\sqrt{3}}{2}\) is \(x \in (- \frac{\pi}{6} + 2k\pi, \frac{\pi}{6} + 2k\pi)\), where \(k\) is an integer.

Step 1: Understand the problem

You are given the inequality \(\cos x > \frac{\sqrt{3}}{2}\). The objective is to solve for \(x\). The range of \(\cos x\) is between -1 and 1 inclusive, and its period is \(2\pi\) (or 360° if working in degrees).

Step 2: Solve cosine inequality

The cosine function attains a value of \(\frac{\sqrt{3}}{2}\) at angles \(x = \frac{\pi}{6}\) and \(x = - \frac{\pi}{6}\) in each period of \(2\pi\). However, we are interested in angles at which the cosine function is greater than \(\frac{\sqrt{3}}{2}\). This occurs for \(x\) in the open intervals \((- \frac{\pi}{6} + 2k\pi, \frac{\pi}{6} + 2k\pi)\), where \(k\) is an integer.

Step 3: Create the solution set

So, the solution set to the inequality \(\cos x > \frac{\sqrt{3}}{2}\) is \(x \in (- \frac{\pi}{6} + 2k\pi, \frac{\pi}{6} + 2k\pi)\) for all \(k\) in integers.

Key Concepts

Cosine Function

The cosine function, commonly denoted as \( \cos x \), is one of the fundamental trigonometric functions. It is often used to determine the horizontal component of a point on a unit circle. The cosine of an angle is defined as the adjacent side over the hypotenuse in a right triangle.

• The range of the cosine function is from -1 to 1, meaning \( \cos x \) can never be greater than 1 or less than -1.
• The cosine is periodic, repeating its values in cycles, which is crucial for solving trigonometric inequalities.

To solve inequalities like \( \cos x > \frac{\sqrt{3}}{2} \), understanding where the cosine function achieves specific values is vital. The cosine function reaches a maximum value of 1 when \( x = 0 \), and values close to it, such as \( \frac{\sqrt{3}}{2} \), can be observed at well-known angles such as \( x = \frac{\pi}{6} \) and \( x = -\frac{\pi}{6} \). These key angles form the basis of solving related trigonometric inequalities effectively.

Periodic Functions

The concept of periodicity is fundamental to understanding the behavior of trigonometric functions, including the cosine function. A function is periodic if it repeats its values at regular intervals. For the cosine function, this interval is also known as its period.

• The period of the cosine function is \( 2\pi \) radians (or 360 degrees).
• This means that every \( 2\pi \) units, the values of the cosine function repeat.

Because of this periodic nature, when solving trigonometric inequalities, such as \( \cos x > \frac{\sqrt{3}}{2} \), we exploit the repeatability of cosine. Once we find solutions within one period, we can easily extend these solutions to any period by adjusting in steps of \( 2k\pi \), where \( k \) is an integer. This is why the solutions mentioned appear as intervals like \((-\frac{\pi}{6} + 2k\pi, \frac{\pi}{6} + 2k\pi)\), allowing us to account for the repeating nature of the cosine function.

Inequality Solutions

Solving trigonometric inequalities involves finding all angle values that satisfy the given condition. In the case of the inequality \( \cos x > \frac{\sqrt{3}}{2} \), the aim is to determine when the cosine function is greater than the specified value.Steps for solving the inequality:1. **Identify Key Angles**: - Determine angles where \( \cos x = \frac{\sqrt{3}}{2} \), which are \( x = \frac{\pi}{6} \) and \( x = -\frac{\pi}{6} \). These are the critical boundaries where the value changes.2. **Establish Interval**: - Recognize where \( \cos x \) exceeds \( \frac{\sqrt{3}}{2} \), which is within \( ( -\frac{\pi}{6}, \frac{\pi}{6} ) \) in one period.3. **Formulate General Solutions**: - Due to periodicity, generalize the solution to all cycles by adjusting with \( 2k\pi \), resulting in intervals like \(( -\frac{\pi}{6} + 2k\pi, \frac{\pi}{6} + 2k\pi)\).By following these steps, you can solve not only this specific inequality but also any similar cosine inequality, understanding both the behavior of the function and exploiting its regularity.