Problem 6 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Given \(\begin{aligned} f(x) &=\frac{\cos x-\sin x}{\cos 2 x}, \quad x \neq \frac{\pi}{4} \\ &=\frac{1}{\sqrt{2}}, \quad x=\frac{\pi}{4} . \end{aligned}\)

Step-by-Step Solution

Answer

In the given piecewise function \(f(x)\), the values for the expression \(x \neq \frac{\pi}{4}\) and \(x = \frac{\pi}{4}\) do not match at \(x = \frac{1}{\sqrt{2}}\). The value for the expression at \(x \neq \frac{\pi}{4}\) for \(x = \frac{\pi}{4}\) is 0, while the expression at \(x = \frac{\pi}{4}\) is \(\frac{1}{\sqrt{2}}\), and 0 is not equal to \(\frac{1}{\sqrt{2}}\). Thus, the given function is incorrect.

Step 1: Calculate the value of the expression for \(x \neq \frac{\pi}{4}\) at \(x = \frac{\pi}{4}\)

When \(x = \frac{\pi}{4}\), we can plug it into the expression for \(f(x)\) when \(x \neq \frac{\pi}{4}\): \[f\left(\frac{\pi}{4}\right) = \frac{\cos\left(\frac{\pi}{4}\right)-\sin\left(\frac{\pi}{4}\right)}{\cos\left(2\left(\frac{\pi}{4}\right)\right)}\] To solve this expression, we apply general trigonometric identities and the properties of the cosine and sine functions.

Step 2: Apply Trigonometric Identities

First, we need to recall the following trigonometric identities 1. \(\cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\) 2. \(\cos(2x) = 1 - 2\sin^2(x)\) Now, we replace the expression in the numerator and the angle in the cosine function with the appropriate values from the identities: \[f\left(\frac{\pi}{4}\right) = \frac{\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}}{\cos(2\left(\frac{\pi}{4}\right))}\] The expression simplifies as: \[f\left(\frac{\pi}{4}\right) = \frac{0}{\cos(\pi)}\] We know that \(\cos(\pi) = -1\). Therefore, \[f\left(\frac{\pi}{4}\right) = \frac{0}{-1} = 0\]

Step 3: Compare the Values

Now we need to check if the value of \(f(x)\) for \(x = \frac{\pi}{4}\) from the given formula (\(\frac{1}{\sqrt{2}}\)) is equal to the value we just computed (\(0\)). In this case, we find: \[\frac{1}{\sqrt{2}}\neq0\] #Conclusion# The values of the function \(f(x)\) at \(x = \frac{\pi}{4}\), when calculated from the two different expressions, are not equal. Therefore, the given function is incorrect.

Key Concepts

Trigonometric Identities

Trigonometric identities are essential formulas that relate the angles and sides of a triangle in trigonometry. They are used to simplify expressions and solve trigonometric equations.
One of the most fundamental identities is the Pythagorean identity:

• \( \sin^2(x) + \cos^2(x) = 1 \)

Other important ones include angle sum and difference formulas, double angle identities, and the half-angle identities. In our exercise, we come across the double angle identity for cosine:

• \( \cos(2x) = \cos^2(x) - \sin^2(x) \)
• \( \cos(2x) = 1 - 2\sin^2(x) \)
• \( \cos(2x) = 2\cos^2(x) - 1 \)

By applying these identities, we simplified the expression in the function to solve a particular case for \( x = \frac{\pi}{4} \). These identities help us reframe trigonometric expressions to make calculations easier, which is vital in both theoretical and applied mathematics.

Piecewise Functions

Piecewise functions are unique types of functions that are defined by different expressions over different intervals. They provide flexibility in mathematical modeling and are used when a function exhibits distinctly different behaviors over different parts of its domain.
Consider the function given in the exercise:

• For \( x eq \frac{\pi}{4} \), the function is defined as \( \frac{\cos x- \sin x}{\cos 2 x} \).
• For \( x = \frac{\pi}{4} \), the function is assigned a constant value \( \frac{1}{\sqrt{2}} \).

Piecewise functions are often used to handle points of discontinuity or to define functions with abrupt changes in behavior. Such functions require careful analysis to ensure they make sense across all their defined pieces. In our exercise, checking the consistency of these pieces at the point of transition \( x = \frac{\pi}{4} \) exposed an inconsistency in the function's definition.

Trigonometric Functions

Trigonometric functions are the building blocks of trigonometry, relating the angles of a right triangle to the ratios of two of its sides. The primary trigonometric functions are sine, cosine, and tangent. We utilize them regularly in calculus, engineering, and many other fields due to their periodic nature.
In our exercise, we primarily deal with cosine and sine:

• \( \cos(x) \) denotes the adjacent side over the hypotenuse in a right triangle.
• \( \sin(x) \) represents the opposite side over the hypotenuse.

These functions also have specific values at standard angles. For example, at \( x = \frac{\pi}{4} \), both sine and cosine equal \( \frac{1}{\sqrt{2}} \). Understanding these standard values is crucial for quickly evaluating trigonometric expressions without having to calculate them anew each time. Our exercise highlights this property when evaluating at \( x = \frac{\pi}{4} \).

Function Evaluation

Function evaluation involves finding the output value of a function for a given input. This is a critical skill in mathematics, as it helps understand the behavior of a function over its domain.
For example, suppose we have a function \( f(x) \). Evaluating \( f(x) \) at \( x = a \) means substituting \( a \) into the function to find \( f(a) \).

• In our exercise, the task involved evaluating \( f \) at \( x = \frac{\pi}{4} \).

It is not just about computing an output but also ensuring all function rules and piecewise definitions are correctly applied. It often involves algebraic manipulation, application of identities, or numerical computation. Moreover, in cases involving piecewise functions, special attention is needed to apply the correct piece that matches the intended input value, ensuring the function's overall correctness and integrity. This process helps identify inconsistencies or errors, as seen in the exercise when we compared outputs from different formulas.