Problem 128 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

\(f(x)=3 \sin \sqrt{\frac{\pi^{2}}{16}-x^{2}}\)

Step-by-Step Solution

Answer

The function \(f(x) = 3 \sin \sqrt{\frac{\pi^{2}}{16} - x^{2}}\) consists of several components: a constant (3), a sine function, a square root, and an expression inside the square root. Each component plays a role in shaping the final function. The constant 3 increases the amplitude of the sine wave, the sine function provides the wave-like shape and periodic property, the square root ensures the function is non-negative and real-valued for defined x-values, and the expression inside square root affects the period and shape of the sine wave.

Step 1: Understand the Function Components

The function \(f(x)\) is composed of several components - a constant, sine function, square root, and an expression inside the square root. Let's try to understand each component: 1. Constant (3): This affects the amplitude of the sine wave, making it three times higher. 2. Sine function: The sine function oscillates between -1 and 1 and is a periodic function. 3. Square root: Takes the non-negative square root of the expression inside. 4. Expression inside square root (\(\frac{\pi^{2}}{16} - x^{2}\)): This expression affects the period and shape of the sine wave.

Step 2: Graphing the Function Components Separately

To better understand how the components of the function contribute to the overall function, we'll graph them separately: 1. Constant (3): The constant does not have a graph in itself, as it only affects the amplitude of the sine function. 2. Sine function: A simple sine function can be graphed as a wave with amplitude 1 and period \(2\pi\). 3. Square root: The square root of a real number is non-negative. It should be noted that if the value inside the square root is negative, the function will be undefined for that input. 4. Expression inside square root: The graph of \(\frac{\pi^{2}}{16} - x^{2}\) is a parabola.

Step 3: Combining the Components

Now that we have a basic understanding of each component, let's break down the function step by step: 1. Start with the expression inside the square root, \(\frac{\pi^2}{16} - x^2\). Substitute this expression into the square root function, obtaining \(\sqrt{\frac{\pi^2}{16} - x^2}\). 2. Take the sine of the result from step 1, resulting in \(\sin \left( \sqrt{\frac{\pi^2}{16} - x^2} \right)\). 3. Finally, multiply the result from step 2 by the constant, 3: \(3 \sin \left( \sqrt{\frac{\pi^2}{16} - x^2} \right)\). This is our final function, \(f(x) = 3 \sin \left( \sqrt{\frac{\pi^2}{16} - x^2} \right)\).

Step 4: Understanding the Importance of Each Component

After going through the steps above, we should now have a better understanding of how each component contributes to the function. In summary: 1. The constant (3) increases the amplitude (height) of the sine wave by three times. 2. The sine function gives the function its wave-like shape with a periodic property. 3. The square root ensures that the function is real-valued and non-negative for defined values of x. 4. The expression inside the square root both affects the period and shape of the sine wave and defines the range of x where the function is real-valued.

Key Concepts

Amplitude

The amplitude of a trigonometric function refers to the height of its peaks and the depth of its troughs. It is essentially how tall the wave is. For the function \( f(x) = 3 \sin \left( \sqrt{\frac{\pi^{2}}{16} - x^{2}} \right) \), the amplitude is determined by the constant multiplier right in front of the sine function, which is 3 in this case.

This means that the peaks of the sine wave can go up to 3, and the troughs can go down to -3. The amplitude affects the vertical stretching or shrinking of the graph. If this number were 1, the function would simply oscillate between -1 and 1. The higher the amplitude, the taller the wave.

• Amplitude is the vertical stretch factor for a sine wave.
• In our function, the amplitude is 3, making the wave three times taller than usual.

Period

The period of a trigonometric function is the distance it takes for the function to complete one full cycle before repeating. A simple sine function, \( \sin(x) \), has a period of \(2\pi\).

In the function \( f(x) = 3 \sin \left( \sqrt{\frac{\pi^{2}}{16} - x^{2}} \right) \), the period becomes more complex due to the presence of the radical expression inside the sine function. The term \( \sqrt{\frac{\pi^{2}}{16} - x^{2}} \) affects how quickly the sine wave completes a cycle. While it's not as straightforward as a standard sine curve's \(2\pi\) period, this complex expression influences the length of one complete wave considerably.

• Period refers to the distance along the x-axis for one complete wave cycle.
• The expression inside the square root can change the effective period of the function.

Square Root Function

The square root function, \( \sqrt{x} \), extracts the principal (non-negative) square root of a number. In our given function, the square root surrounds the expression \( \frac{\pi^2}{16} - x^2 \), which modifies the input values before they are applied to the sine function.

An important aspect of using a square root function is ensuring the expression inside is non-negative, because square roots of negative numbers are not real. This means \( x \) must be in the range where \( \frac{\pi^2}{16} - x^2 \geq 0 \).

• The square root ensures the expression yields non-negative outputs.
• It is crucial for defining the real domain of the function.

Real-Valued Function

A real-valued function is one that produces real numbers as outputs for all inputs in its domain. For \( f(x) = 3 \sin \left( \sqrt{\frac{\pi^{2}}{16} - x^{2}} \right) \), this means we need to find all \( x \) values that ensure the function outputs are real numbers.

The inclusion of the square root function imposes a condition that the expression \( \frac{\pi^2}{16} - x^2 \) must be non-negative to produce real values. The sine function naturally outputs real numbers, but only when its argument is real.

• A real-valued function only deals in real numbers for both input and output.
• The domain of the function must ensure all components are within their real number ranges.

Sine Function

The sine function is a fundamental trigonometric function that describes a smooth periodic wave. It is defined for all real numbers \( x \), cycling between -1 and 1 in value.

In our function \( f(x) = 3 \sin \left( \sqrt{\frac{\pi^{2}}{16} - x^{2}} \right) \), the sine function takes the result of \( \sqrt{\frac{\pi^{2}}{16} - x^{2}} \) as its input. This process combines to form a wave that retains the periodic nature of the sine function, albeit in a more complicated form due to the transform.

• The sine function describes cyclical oscillations based on angles.
• It is essential in generating the wave-like behavior of the full function.