Problem 13 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(f(x)=\sin \left[\pi^{2}\right] x+\sin \left[-\pi^{2}\right] x\), where [ \(]\) denotes the greatest integer function, then (A) \(f\left(\frac{\pi}{2}\right)=1\) (B) \(f(\pi)=2\) (C) \(f\left(\frac{\pi}{4}\right)=-1\) (D) None of these

Step-by-Step Solution

Answer

(D) None of these.

Step 1: Evaluate the Greatest Integer Function

First, simplify the expression given in the problem: \( \sin[\pi^2]x + \sin[-\pi^2]x \). Since \([x]\) denotes the greatest integer less than or equal to \(x\), evaluate \(\pi^2 \approx 9.8696\). This means \([\pi^2] = 9\) and \([-\pi^2] = -10\). Therefore, the expression simplifies to \(\sin(9x) + \sin(-10x)\).

Step 2: Simplify Using Trigonometric Identities

Utilize the identity \(\sin(-x) = -\sin(x)\) to further simplify the expression: \(\sin(9x) - \sin(10x)\).

Step 3: Use Angle Difference Identity

Apply the identity for the difference of sines, \(\sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)\), to the expression \(\sin(9x) - \sin(10x)\). This results in:\[ 2 \cos\left(\frac{9x + 10x}{2}\right) \sin\left(\frac{9x - 10x}{2}\right) = 2 \cos\left(\frac{19x}{2}\right) \sin\left(-\frac{x}{2}\right)\].

Step 4: Evaluate Given Options

Evaluate \( f\left(\frac{\pi}{2}\right) \), \( f(\pi) \), and \( f\left(\frac{\pi}{4}\right) \) using the derived expression. For option (A), substitute \(x = \frac{\pi}{2}\) into the expression:\[ 2 \cos\left(\frac{19\pi}{4}\right) \sin\left(-\frac{\pi}{4}\right) = 2 \cos\left(\frac{19\pi}{4}\right) \left(-\frac{\sqrt{2}}{2}\right) \]which does not equal 1;For (B), substitute \(x = \pi\):\[ 2 \cos\left(\frac{19\pi}{2}\right) \sin\left(-\frac{\pi}{2}\right) = 2 \times 0 \times \left(-1\right) = 0 \]which is not 2;For (C), substitute \(x = \frac{\pi}{4}\):\[ 2 \cos\left(\frac{19\pi}{8}\right) \sin\left(-\frac{\pi}{8}\right) \]which cannot be simplified to -1. None of the given options match the evaluated values.

Key Concepts

Greatest Integer Function

The greatest integer function, often denoted as \([x]\), is a mathematical concept that assigns any given number the largest integer less than or equal to it. This function is particularly useful when dealing with decimal numbers, as it effectively "rounds down" to the nearest whole number.

• For example, \([3.7] = 3\).
• Similarly, \([-2.4] = -3\), since \(-3\) is less than \(-2.4\).

In our problem, we applied the greatest integer function to \(\pi^2\), where \(\pi^2 \approx 9.8696\). This evaluates to \([\pi^2] = 9\) and \([-\pi^2] = -10\).
These values were pivotal in simplifying the expression \(\sin(9x) + \sin(-10x)\). Understanding how the greatest integer function works helps make complex trigonometric expressions more manageable.

Angle Difference Identity

The angle difference identity is a key tool in trigonometry, enabling us to simplify and manipulate expressions that involve the difference of angles. It is especially useful in deriving formulas for trigonometric functions such as sine and cosine.

In this exercise, we focused on the identity for the sine of angle differences:

• \( \sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \)

To apply this, we substitute the values \(A = 9x\) and \(B = 10x\), leading us to the expression \(2 \cos\left(\frac{19x}{2}\right) \sin\left(-\frac{x}{2}\right)\).
This simplification is crucial as it reduces the original expression into a form that's easier to evaluate with specific angle values.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are universally true for all values of the involved variables. These identities are essential tools in simplifying and solving trigonometric expressions and equations.

In this problem, the following identities were critical:

• The sine identity, \( \sin(-x) = -\sin(x) \), helped transform \(\sin(-10x)\) to \(-\sin(10x)\), reshaping the original expression.
• The angle difference identity, as previously discussed, further simplified the function.

Applying these trigonometric identities allows us to navigate complex functions effectively, providing clarity and structure to what might otherwise appear to be intricate problems. Mastering these identities gives us a robust foundation for understanding and solving trigonometric equations.