Problem 2 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

\(\cos (2 x-3)<0\)

Step-by-Step Solution

Answer

The solution to the inequality \(\cos (2x - 3) < 0\) will be \(x > ((π/2 + α +3) / 2)\) and \(x < ((3π/2 + α + 3) / 2)\), for integer \(n\).

Step 1: Identify the intervals where cosine is negative

Recall that cosine function is negative in the second \((π/2 < x < π)\) and third quadrants \((π < x < 3π/2)\). Therefore, \(\cos (x) < 0\) for \(π/2 < x < 3π/2\). Adding any full period of \(2π\) to these intervals will also give intervals where the cosine is negative.

Step 2: Adjust for the transformed function

The given function is not simply \(\cos (x)\), but \(\cos (2x - 3)\). This represents a shift and rescaling of the input value. To isolate the input to the cosine function, set \(2x - 3 = α\). So our intervals for negative cosine values become \(π/2 + α < x < 3π/2 + α\). We need to find the values of \(α\) that fall within these intervals.

Step 3: Solve for x

In the modified form \(2x - 3 = α\), solve for \(x\). The solutions will be \(x > ((π/2 + α +3) / 2)\) and \(x < ((3π/2 + α + 3) / 2)\), for integer \(n\). These are the intervals for which \(\cos (2x - 3) < 0\).

Key Concepts

Cosine Function

The cosine function is one of the fundamental trigonometric functions. It is often denoted as \( \cos(x) \), where \( x \) is the angle in radians. The cosine function relates to the x-coordinate of a point on the unit circle.

• Range: The range of \( \cos(x) \) is between -1 and 1.
• Symmetry: Cosine is an even function. This means \( \cos(-x) = \cos(x) \).
• Behavior: It starts from 1 at \( x=0 \), dips to -1, and comes back to 1 again as \( x \) goes from 0 to \( 2\pi \).

For this exercise, we focus on when \( \cos(x) \) is negative. This occurs when the angle corresponds to specific parts of the unit circle. Understanding cosine's properties helps us predict its behavior over different intervals.

Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four quadrants. Each quadrant influences the sign of trigonometric functions.

• Quadrant I: All trigonometric functions are positive.
• Quadrant II: Sine is positive, but cosine and tangent are negative.
• Quadrant III: Tangent is positive, but sine and cosine are negative.
• Quadrant IV: Cosine is positive, while sine and tangent are negative.

The cosine function is negative in the second and third quadrants. For the inequality \( \cos(2x - 3) < 0 \), we focus on these quadrants. The angles in these quadrants align with where the cosine is less than zero. Interpreting which quadrant the angle is in helps solve the inequality.

Periodic Functions

The cosine function is a periodic function, meaning it repeats its values in regular intervals. This periodic nature is crucial for understanding trigonometric inequalities.

• Period: The period of \( \cos(x) \) is \( 2\pi \).
• Repetition: This means every \( 2\pi \) interval, the cosine pattern repeats itself.
• Applications: In equations like \( \cos(2x - 3) < 0 \), adjusting for transformations such as stretching or shifting can change the intervals where cosine is negative.

When solving inequalities with periodic functions, such as this exercise, the key is to account for these repeated intervals. Whether dealing with transformations or shifts, understanding the period helps predict where the function will repeat specific values. This knowledge allows us to find all solutions within a given domain.