Problem 150 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The period of \(\sin ^{2} \theta\) is : \(\quad\) (A) \(\pi^{2}\) (B) \(\pi\) (C) \(2 \pi\) (D) \(\pi / 2\)

Step-by-Step Solution

Answer

The period of \( \sin^2 \theta \) is \( \pi \) (option B).

Step 1: Understand the Problem

We need to find the period of the function \( \sin^2 \theta \). The period of a function is the smallest positive value \( P \) such that the function repeats itself, i.e., \( f(\theta + P) = f(\theta) \) for all \( \theta \).

Step 2: Analyze \( \sin \theta \)

The basic sine function \( \sin \theta \) has a period of \( 2\pi \). This means \( \sin(\theta + 2\pi) = \sin(\theta) \).

Step 3: Apply Squaring Transformation

When we square the sine function to get \( \sin^2 \theta \), the new function becomes \( \sin^2(\theta + 2\pi) = (\sin(\theta + 2\pi))^2 = \sin^2(\theta) \).

Step 4: Simplify Using Trigonometric Identity

Recall the identity: \( \sin^2 \theta = \frac{1 - \cos(2\theta)}{2} \). The cosine function \( \cos(2\theta) \) has a period of \( \pi \) because \( \cos(2(\theta + \pi)) = \cos(2\theta + 2\pi) = \cos(2\theta) \).

Step 5: Identify the Period of \( \sin^2 \theta \)

With the transformation \( \sin^2 \theta = \frac{1 - \cos(2\theta)}{2} \), the period of \( \sin^2 \theta \) is determined by the cosine term with period \( \pi \). Thus, the smallest period is \( \pi \).

Step 6: Verify the Period

Verify that moving from \( \theta \) to \( \theta + \pi \) gives \( \sin^2(\theta + \pi) = \sin^2 \theta \). This confirms that the period is indeed \( \pi \).

Key Concepts

Sine Function

The sine function is one of the basic trigonometric functions and plays a crucial role in various mathematical contexts. Its primary characteristic is its periodic behavior. A standard sine function can be described by the equation \( \sin \theta \), which represents the vertical component of a point on the unit circle.

• The sine function oscillates between -1 and 1 as \( \theta \) changes.
• The period of the sine function is \( 2\pi \), meaning after every \( 2\pi \) radians, the sine wave repeats its pattern.
• Key angles such as 0, \( \pi/2 \), \( \pi \), and \( 3\pi/2 \) mark critical points where the sine function hits its maximum, minimum, or passes through the x-axis.

Understanding this wave-like nature and its periodicity is essential when dealing with transformed sine expressions such as \( \sin^2 \theta \). Here, the sine function's properties help determine the behavior of more complex expressions.

Cosine Function

The cosine function complements the sine function in many ways. Like the sine function, it is fundamental in trigonometry and shares periodic properties. The cosine of an angle \( \cos \theta \) represents the horizontal component of a point on the unit circle.

• It oscillates between -1 and 1, just like the sine function.
• The cosine function's period is also \( 2\pi \), signifying that \( \cos(\theta + 2\pi) = \cos \theta \).
• Notably, \( \cos \theta \) equals 1 when \( \theta = 0, 2\pi, 4\pi \), and so forth, and equals -1 at \( \theta = \pi, 3\pi, 5\pi \), etc.

A unique aspect of the cosine function is how it's used in conjunction with sine to form identities, such as in the squared sine identity \( \sin^2 \theta = \frac{1 - \cos(2\theta)}{2} \). This specific identity is significant when determining the period of \( \sin^2 \theta \).

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all angles. These identities play a crucial role in simplifying trigonometric expressions and solving related problems.

• One of the classic identities is \( \sin^2 \theta + \cos^2 \theta = 1 \). This identity is the basis for deriving other useful formulas.
• For our specific problem, the identity \( \sin^2 \theta = \frac{1 - \cos(2\theta)}{2} \) is particularly important. It allows us to understand \( \sin^2 \theta \) in terms of a cosine function.
• Using this identity, the period of \( \sin^2 \theta \) becomes apparent because \( \cos(2\theta) \) has a period of \( \pi \).

These identities not only simplify computations but also provide deeper insights into periodic functions. They help unravel complexities in trigonometric functions, making seemingly difficult problems manageable.