Problem 17 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

Period of the function \(y=\sin \frac{2 t+3}{6 \pi}\) is (a) \(3 \pi^{2}\) (b) \(5 \pi^{2}\) (c) \(7 \pi^{2}\) (d) \(6 \pi^{2}\)

Step-by-Step Solution

Answer

The period of the function is \(6\pi^2\), so the answer is option (d).

Step 1: Identify the Standard Form of the Sine Function

The standard form of a sine function is given by \( y = \sin(x) \), where the period is \(2\pi\). For a function of the form \( y = \sin(bx) \), the period is given by \(\frac{2\pi}{b}\).

Step 2: Identify the Argument of the Given Function

In the given function, \( y = \sin\left(\frac{2t + 3}{6\pi}\right) \), the argument of the sine function is \( x = \frac{2t + 3}{6\pi} \).

Step 3: Determine the Multiplier for the Period Calculation

Rewrite the argument as \( \frac{1}{6\pi} \cdot (2t + 3) \). The coefficient of \( t \) is \( \frac{1}{6\pi} \cdot 2 = \frac{2}{6\pi} = \frac{1}{3\pi} \). This is the multiplier, \( b \), used for adjusting the period.

Step 4: Calculate the Period of the Function

Given the multiplier \( b = \frac{1}{3\pi} \), the period \( T \) of the sine function is \( \frac{2\pi}{b} = \frac{2\pi}{\frac{1}{3\pi}} \).

Step 5: Simplify the Expression for the Period

Calculate \( \frac{2\pi}{\frac{1}{3\pi}} \) by multiplying by the reciprocal: \( 2\pi \cdot \frac{3\pi}{1} = 6\pi^{2} \).

Step 6: Match the Calculated Period to the Given Options

The calculated period is \( 6\pi^2 \), which matches option (d).

Key Concepts

Understanding the Sine Function

The sine function is one of the fundamental trigonometric functions that helps us describe waves and oscillations. It is defined as the y-coordinate of a point on the unit circle in the Cartesian coordinate system, corresponding to a specific angle from the positive x-axis. When you graph the sine function, it looks like a continuous wave moving above and below the horizontal axis.

The standard sine function is written as \( y = \sin(x) \), where \( x \) represents the angle in radians. This function has several key characteristics:

• The amplitude, which is the peak value of the wave, is 1.
• The period, the length required for the function to complete one full cycle, is \( 2\pi \).
• The function is symmetric about the origin, indicating an odd function property: \( \sin(-x) = -\sin(x) \).

These properties make the sine function extremely useful not only in mathematics but also in physics and engineering for modeling periodic phenomena such as sound waves and light cycles.

Period Calculation of Trigonometric Functions

The period of a trigonometric function like sine is critical in determining how often the function repeats itself. For a basic sine function, the period is \( 2\pi \), but this period can change if the function is transformed by multiplying the angle inside the sine function by a number. This number, denoted as \( b \), changes the frequency of the wave, which inversely affects the period.

In the form \( y = \sin(bx) \), the period \( T \) of the sine function is given by the formula:

• \( T = \frac{2\pi}{b} \)

In this context, \( b \) is the frequency, and the period is the reciprocal of frequency times \( 2\pi \).
To calculate the period of our given function \( y = \sin\left(\frac{2t + 3}{6\pi}\right) \), the transformed coefficient was \( \frac{1}{3\pi} \). Applying the period formula:

• \( T = \frac{2\pi}{\frac{1}{3\pi}} \)

This results in the period \( 6\pi^2 \), as we multiply by the reciprocal.

Exploring Trigonometric Identities

Trigonometric identities are equations that are always true for any angle, making them essential tools in trigonometry and geometry. They allow us to simplify expressions, solve trigonometric equations, and understand the relationships between various trigonometric functions.

Some fundamental trigonometric identities include:

• Pythagorean Identity: \( \sin^2(x) + \cos^2(x) = 1 \)
• Angle Sum and Difference Identities:
  • \( \sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b) \)
  • \( \sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b) \)
• Double Angle Identity: \( \sin(2x) = 2\sin(x)\cos(x) \)

These identities reflect the deep connections between trigonometric functions and are crucial for solving complex problems. By recognizing and applying these identities, calculations become more straightforward, as illustrated in the period calculation of transformed trigonometric functions.