Problem 212 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \left.\cos x<\frac{\sqrt{3}}{2} \text { \\{Ans. }\left(2 n \pi+\frac{\pi}{6}, 2 n \pi+\frac{11 \pi}{6}\right)\right\\} $$

Step-by-Step Solution

Answer

The solution to the inequality \(\cos(x) < \frac{\sqrt{3}}{2}\) is \(x\) in the set {(2nπ+π/6, 2nπ+5π/6) and (2nπ+7π/6, 2nπ+11π/6)}, where \(n\) is any integer.

Step 1: Identify the range of the cosine function

The cosine function has a range of \([-1,1]\). This means that the inequality \(\cos(x) < \frac{\sqrt{3}}{2}\) has potential solutions for any real number \(x\).

Step 2: Analyze cosine on the unit circle

On the unit circle, \(\cos(x)\) corresponds to the x-coordinate. Observing the unit circle, we see that for every rotation of \(2\pi\), \(\cos(x)\) is less than \(\frac{\sqrt{3}}{2}\) between \(\frac{\pi}{6}\) and \(\frac{5\pi}{6}\), and between \(\frac{7\pi}{6}\) and \(\frac{11\pi}{6}\).

Step 3: Define the solution set for the inequality

Combining the solutions from the two intervals in each rotation of the unit circle, the inequality \(\cos(x) < \frac{\sqrt{3}}{2}\) is satisfied for \(x\) in the set {(2nπ+π/6, 2nπ+5π/6) and (2nπ+7π/6, 2nπ+11π/6)}, where \(n\) is any integer.

Key Concepts

Unit Circle

The unit circle is a fundamental concept in trigonometry. It is a circle with a radius of 1, centered at the origin of a coordinate plane. The unit circle simplifies the understanding of trigonometric functions because any point on the circle can be used to determine the values of sine and cosine for various angles.
In the unit circle, an angle \( x \) (measured in radians) can be defined as the measure of the angle formed with the positive x-axis. The corresponding point on the circle is \( (\cos(x), \sin(x)) \). Here, \( \cos(x) \) is the x-coordinate, and \( \sin(x) \) is the y-coordinate.

• The circle's coordinates wrap periodically with a full rotation being \( 2\pi \) radians.
• Angles can continue beyond \( 2\pi \) radians, maintaining periodic behavior.

Understanding positions of angles on the unit circle is crucial for solving trigonometric inequalities since they help identify the sign and magnitude of sine and cosine at various rotations.

Cosine Function

The cosine function, denoted as \( \cos \), relates an angle in a right triangle to the ratio of the length of the adjacent side over the hypotenuse. On the unit circle, \( \cos(x) \) is the x-coordinate of a point as already mentioned, which vividly illustrates that cosine values range from \(-1\) to \(1\).
Different angles correspond to different x-coordinates on the unit circle, translating to distinct cosine values:

• \( \cos(0) = 1 \)
• \( \cos(\pi/2) = 0 \)
• \( \cos(\pi) = -1 \)
• periodically repeated for \( 2\pi \)

Since the cosine function is periodic, it repeats every \( 2\pi \) radians, making it crucial for understanding how solutions to trigonometric inequalities like \( \cos(x) < \frac{\sqrt{3}}{2} \) work. Observing when \( \cos(x) \) achieves different values on the unit circle helps form the intervals where the inequality holds.

Inequalities

An inequality is a mathematical statement that one quantity is larger or smaller than another. In trigonometry, we often deal with inequalities that include trigonometric functions such as cosine. For example, determining when \( \cos(x) < \frac{\sqrt{3}}{2} \) implies finding the range of angles \( x \) for which this inequality is true.
Looking at the unit circle:

• When \( x \) is in the interval \( \left( \frac{\pi}{6}, \frac{5\pi}{6} \right) \), the cosine values drop below \( \frac{\sqrt{3}}{2} \).
• The same happens between \( \left( \frac{7\pi}{6}, \frac{11\pi}{6} \right) \).

These intervals are crucial to solving the inequality as they give all possible \( x \) values through any full rotation of the unit circle (which adds an integer multiple of \( 2\pi \) to \( x \)). By recognizing this repeating pattern of intervals around the circle, we can smartly derive the solution set of the inequality for all real numbers \( x \).