Problem 8 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

\(4 \cos ^{2} x-3 \geq 0\)

Step-by-Step Solution

Answer

The solution for the inequality \(4 \cos ^{2} x-3 \geq 0\) is \(0 \leq x \leq \frac{\pi}{6}\) and \(\frac{5\pi}{6} \leq x \leq 2\pi\).

Step 1: Rewrite The Inequality

The quadratic inequality to solve is \(4 \cos ^{2} x-3 \geq 0\). It can be rewritten in terms of cosine: \(\cos ^{2} x \geq \frac{3}{4}\).

Step 2: Find The Values of Cosine

For inequality \(\cos ^{2} x \geq \frac{3}{4}\), take the square root of both sides which result in: \(|\cos x| \geq \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}\).

Step 3: Determine The Interval within Given Period

Our solutions for \(x\) values must be in the interval \([0, 2\pi]\). Therefore, the values of \(x\) are determined by \(0 \leq x \leq \frac{\pi}{6}\) and \(\frac{5\pi}{6} \leq x \leq 2\pi\). This is because the function \(\cos x\) is equal to \(\frac{\sqrt{3}}{2}\) at \(x = \frac{\pi}{6}\) and \(x = \frac{5\pi}{6}\), and it is more than \(\frac{\sqrt{3}}{2}\) for \(0 \leq x < \frac{\pi}{6}\) and \(\frac{5\pi}{6} < x \leq 2\pi\).

Key Concepts

Cosine Function

To understand the trigonometric inequality involving the cosine function, it's important to grasp what the cosine function itself represents. Cosine is a fundamental trigonometric function that relates to the angle in a right triangle.

• The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.
• In the unit circle, cosine represents the horizontal distance from the origin to a point on the circle.
• Cosine values range from -1 to 1, as they dictate positions along the circle's circumference.

In problems involving cosine, such as inequalities, we often seek specific angle ranges where cosine takes or exceeds certain values.
In the given exercise, we deal with a term \(\cos^2 x\), indicating that the cosine values are squared and compared in the inequality. Calculating this helps identify possible angle ranges satisfying the condition.

Quadratic Inequality

Quadratic inequalities, like the one we have \(4 \cos^2 x - 3 \geq 0\), can be thought of like solving quadratic equations, but with a condition of inequality.

• The term \(4 \cos^2 x\) indicates it is a quadratic expression in terms of \(\cos x\).
• We simplify quadratic inequalities by rearranging them, isolating terms, and then solving for the variable, as done in the solution to \(\cos^2 x \geq \frac{3}{4}\).
• Taking the square root of both sides provides two possible conditions, \(|\cos x| \geq \frac{\sqrt{3}}{2}\).

This step is crucial in trigonometric inequalities, allowing us to translate a seemingly complex expression into something manageable: knowing when cosine is greater than or equal to a specific value.
Breaking it into smaller, familiar parts helps achieve the condition needed for defining angles of interest.

Trigonometric Solutions

Finding the solutions to trigonometric equations involves understanding the behavior of the trigonometric functions over a certain interval. With cosine, this behavior over a period gives significant insight. Once you have established the bound as in \(|\cos x| \geq \frac{\sqrt{3}}{2}\), finding angles that satisfy this becomes a matter of understanding the decrease and increase pattern of cosine.

• Cosine achieves its maximum (1) at multiple angles within a period, including \(0\) and \(2\pi\).
• For the mathematical range known as \([0, 2\pi]\), \(\cos x = \frac{\sqrt{3}}{2}\) at \(x = \frac{\pi}{6}, \frac{5\pi}{6}\).
• The range is thus divided into two intervals based on these critical points, which encompass all the angles for which \(\cos x\) satisfies the inequality.

This breakdown shows how cosine's periodic nature aids in determining regions for specific inequations like the one given, ensuring that all values meeting the criteria are captured effectively.