Problem 11 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

Period of \(\sin \left(2 \pi x+\frac{\pi}{3}\right)+2 \sin \left(3 \pi x+\frac{\pi}{4}\right)+\) \(3 \sin 5 \pi x\) is (a) 1 (b) \(2 / 3\) (c) \(2 / 5\) (d) 2

Step-by-Step Solution

Answer

The period is 2.

Step 1: Identify Frequencies

The function given is a sum of three sine waves. The argument of each sine wave will determine its frequency:1. For \(\sin(2\pi x + \frac{\pi}{3})\), the frequency is \(2\pi\).2. For \(2\sin(3\pi x + \frac{\pi}{4})\), the frequency is \(3\pi\).3. For \(3\sin(5\pi x)\), the frequency is \(5\pi\).

Step 2: Find Individual Periods

The period of a sine function \(\sin(kx)\) is \(\frac{2\pi}{k}\). Therefore, calculate the period for each component:1. For \(\sin(2\pi x + \frac{\pi}{3})\), the period is \(\frac{2\pi}{2\pi} = 1\).2. For \(2\sin(3\pi x + \frac{\pi}{4})\), the period is \(\frac{2\pi}{3\pi} = \frac{2}{3}\).3. For \(3\sin(5\pi x)\), the period is \(\frac{2\pi}{5\pi} = \frac{2}{5}\).

Step 3: Determine Least Common Multiple

The overall period of the function is the least common multiple (LCM) of the individual periods \(1\), \(\frac{2}{3}\), and \(\frac{2}{5}\). To find the LCM:1. Convert all periods to fractions with a common denominator. The denominators are \(1\), \(3\), and \(5\). The LCM of \(1\), \(3\), and \(5\) is \(15\).2. Convert periods: - \(1 = \frac{15}{15}\)- \(\frac{2}{3} = \frac{10}{15}\)- \(\frac{2}{5} = \frac{6}{15}\)3. The LCM of \(\frac{15}{15}\), \(\frac{10}{15}\), and \(\frac{6}{15}\) is \(\frac{30}{15}\) or \(2\).

Step 4: Conclusion

The least common multiple of the individual periods \(1\), \(\frac{2}{3}\), and \(\frac{2}{5}\) is \(2\). Hence, the period of the given function \(\sin(2\pi x + \frac{\pi}{3}) + 2\sin(3\pi x + \frac{\pi}{4}) + 3\sin(5\pi x)\) is \(2\).

Key Concepts

Frequency of Sine Function

When you come across a sine function like \(\sin(kx)\), the frequency is essential to understand how fast the wave oscillates. It tells us how many cycles the wave completes in a 2\(\pi\) unit interval on the x-axis.
Here's how to determine it:

• Identify the coefficient \(k\) in front of \(x\) within the sine function.
• The frequency of this sine function is \(k\).

For \(\sin(2\pi x + \frac{\pi}{3})\), the frequency is simply \(2\pi\) because the coefficient in front of \(x\) is \(2\pi\).
Similarly, for \(2\sin(3\pi x + \frac{\pi}{4})\), the frequency is \(3\pi\), and for \(3\sin(5\pi x)\), it is \(5\pi\).
This increased coefficient leads to more oscillations over a set interval but does not affect the overall amplitude of the wave.

Least Common Multiple

The Least Common Multiple (LCM) is a term often used in mathematics to find a common period among different functions. This is especially useful when dealing with sums of periodic functions
like our current problem.
To find the LCM:

• Start by converting all periods to fractions with a common denominator.
• The denominators in our example are \(1\), \(3\), and \(5\).
• The LCM of these numbers is \(15\).

Next, you'll need to convert each original period:

• \(1\) becomes \(\frac{15}{15}\)
• \(\frac{2}{3}\) transforms into \(\frac{10}{15}\)
• \(\frac{2}{5}\) becomes \(\frac{6}{15}\)

Finally, finding the LCM of these fractions' numerators guides us to the period of the sum of functions. In our case, \(\frac{30}{15}\), which simplifies to \(2\). This technique ensures that the periods are aligned, providing the simplest repeating cycle for the entire function.

Periodic Functions in Calculus

In calculus, periodic functions play a significant role, especially in trigonometry. Periodic functions, like sine and cosine, repeat their values over a consistent interval known as the period.
Some features include:

• A repeating cycle of values after a defined interval.
• Common examples include the trigonometric functions \(\sin\) and \(\cos\).

For instance, the basic sine function \(\sin(x)\) naturally repeats every \(2\pi\), meaning its period is \(2\pi\). However, when the function argument changes, as it does with coefficients, the period might change too.
This adjustment occurs because the coefficient alters the frequency of oscillations.
Calculus often uses these periodic functions to analyze patterns and solve differential equations, making understanding their periods critical.
By determining their collective period using techniques like finding the LCM of individual periods, calculus transforms into a tool for efficiently analyzing wave patterns and signals.