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

Question

\(|\sin x|>|\cos x|\)

Step-by-Step Solution

Answer

The solution for the inequality \(|\sin x| > |\cos x|\) is when \(x\) lies in the interval \((0, \pi/2)\) and \((3\pi/2, 2\pi)\).

Step 1: Understand the inequality

This inequality says that the absolute value of \(\sin(x)\) is greater than the absolute value of \(\cos(x)\). We have to find those values of \(x\) which satisfy this inequality.

Step 2: Consider each quadrant separately

Remember that \(sin(x)\) has positive values in the first and second quadrants, and negative in the third and the fourth. At the same time, \(\cos(x)\) is positive in the first and the fourth quadrants, and negative in the second and third. Thus, we consider all quadrants separately.

Step 3: Quadrant 1 and Quadrant 3

In the first and third quadrant, both \(\sin(x)\) and \(\cos(x)\) are positive or both are negative. So, inequality \(|\sin x| > |\cos x|\) simplifies to \(\sin x > \cos x\). This holds true when \(x\) lies in the interval \((0, \pi/4)\) and \((5\pi/4, 2\pi)\).

Step 4: Quadrant 2 and Quadrant 4

In the second and fourth quadrant, one of \(\sin(x)\) and \(\cos(x)\) are positive and other is negative. So, inequality \(|\sin x| > |\cos x|\) simplifies to \(-\sin x > \cos x\). This holds true when \(x\) lies in the interval \((\pi/4, \pi/2)\) and \((3\pi/2, 5\pi/4)\).

Step 5: Combine the solutions

Combine the solutions from Step 3 and Step 4. Thus the solution for the inequality \(|\sin x| > |\cos x|\) is when \(x\) lies in the interval \((0, \pi/2)\) and \((3\pi/2, 2\pi)\).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics and have significant applications in various fields. At their core, they describe relationships in right-angled triangles. The primary trigonometric functions—sine, cosine, and tangent—are derived from the ratios of the sides of a triangle.

• Sine (\( ext{sin} x\)): In a right triangle, it is the ratio of the length of the opposite side to the hypotenuse.
• Cosine (\( ext{cos} x\)): This is the ratio of the adjacent side's length to the hypotenuse.
• Tangent (\( ext{tan} x\)): A ratio of the opposite side to the adjacent side.

Each function has a specific range of values dependent on the angle \(x\). These functions are periodic, repeating their values in cycles. For instance, sine and cosine functions repeat every \(2data rendered'}\pi\). Understanding these functions helps in solving more complex trigonometric equations as seen with the inequality \(|\sin x| > |\cos x|\).

Unit Circle

The unit circle is a crucial concept in trigonometry that helps us understand the behavior of trigonometric functions across all angles. It is a circle with a radius of one, centered at the origin \((0,0)\) on a coordinate plane.
By projecting lines from the origin to the circle's circumference, we can define angles and analyze the sine and cosine for those angles.

• Angles are usually measured in radians on the unit circle.
• The coordinates of any point on the unit circle can be written as \((\cos x, \sin x\)).
• The unit circle allows us to define sine and cosine beyond the confines of right triangles, extending to any angle, whether acute or obtuse.

The unit circle also assists in understanding geometrical interpretations of trigonometric inequality like \(|\sin x| > |\cos x|\), by looking at how the projections (and consequently the absolute values) of those functions change as we travel around the circle.

Quadrants

The Cartesian plane divided into four quadrants is essential for understanding where sine and cosine have positive or negative values. These quadrants are numbered counterclockwise starting from the positive x-axis as follows:

• Quadrant I: Both sine and cosine are positive. This is where \(0 < x < \pi/2 \).
• Quadrant II: Sine is positive, cosine is negative. Here, \(\pi/2 < x < \pi \).
• Quadrant III: Both sine and cosine are negative. In this quadrant, \(\pi < x < 3data rendered'}\pi/2\).
• Quadrant IV: Sine is negative, cosine is positive. This quadrant also holds where \(3data rendered'}\pi/2 < x < 2data rendered'}\pi \).

This understanding is applied when solving inequalities like \(|\sin x| > |\cos x|\), as the behavior of these functions is dependent on which quadrant the angle \(x\) is located.

Sine and Cosine Comparison

Comparing sine and cosine functions involves analyzing their values across different sections of the unit circle to solve inequalities like \(|\sin x| > |\cos x|\). The task becomes to determine where on the circle one function dominates the other in magnitude.

• Equality Status: Sine equals cosine at specific points like \(\pi/4\) and \(5data rendered'}\pi/4\), where both functions are \(\frac{\sqrt{2}}{2}\).
• Magnitude Comparison: Away from these points, one function will increase or decrease its absolute value faster than the other.

In quadrants where both are positive or negative, the magnitudes of sine and cosine directly compare. However, when one is negative, their absolute values provide different interpretations of their comparative behavior.
For instance, within \((0, \pi/2)\) and \((3data rendered'}\pi/2, 2data rendered'}\pi)\), the inequality is satisfied, offering essential insight into the behavior of these functions when solving trigonometric problems.