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

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding the Function

To find the period of \(\sin^2 \theta\), we first need to understand that it is derived from the function \(\sin \theta\). The function \(\sin \theta\) has a period of \(2\pi\).

Step 2: Using the Double Angle Formula

The function \(\sin^2 \theta\) can be transformed using the double angle formula: \(\sin^2 \theta = \frac{1 - \cos(2\theta)}{2}\). By expressing \(\sin^2 \theta\) this way, we define it in terms of \(\cos(2\theta)\).

Step 3: Determining the Period of the Transformed Function

Since \(\cos(2\theta)\) is inside the expression, its period is half of the usual cosine function. Typically, \(\cos(\theta)\) has a period of \(2\pi\), but \(\cos(2\theta)\) has a period of \(\frac{2\pi}{2} = \pi\).

Step 4: Relating Back to the Original Function

Because the function \(\sin^2 \theta\) can be described as dependent on \(\cos(2\theta)\), it inherits this period \(\pi\) directly. Hence, the period of \(\sin^2 \theta\) is also \(\pi\).

Key Concepts

Trigonometric Identities

Trigonometric identities are a set of equalities involving trigonometric functions that are true for every value of the variable. These identities are foundational in simplifying expressions and solving trigonometric equations. Some common trigonometric identities include:

• Pythagorean Identity: \( \sin^2 \theta + \cos^2 \theta = 1 \)
• Sum and Difference Formulas: These formulas allow us to express the sine and cosine of a sum or difference of angles in terms of products.
  • \( \sin(a \pm b) = \sin a \cos b \pm \cos a \sin b \)
  • \( \cos(a \pm b) = \cos a \cos b \mp \sin a \sin b \)
• Even-Odd Identities: These dictate that \( \sin(-\theta) = -\sin \theta \) and \( \cos(-\theta) = \cos \theta \).

Understanding these identities can help simplify complex trigonometric expressions, like transforming \( \sin^2 \theta \) using its double angle identity. This simplification aids in determining properties like periodicity more easily.

Double Angle Formula

The double angle formulas are specific trigonometric identities that express trigonometric functions of double angles in terms of single angles. They are crucial when manipulating expressions like \( \sin^2 \theta \).For example, the double angle formula for sine is:\[\sin(2\theta) = 2 \sin \theta \cos \theta\]For cosine, the double angle is expressed as:\[\cos(2\theta) = \cos^2 \theta - \sin^2 \theta\]An alternative form related directly to the expression we're examining is:\[\sin^2 \theta = \frac{1 - \cos(2\theta)}{2}\]This version is particularly useful because it writes \( \sin^2 \theta \) in terms of a single cosine function. The transformation simplifies the analysis of periodicity, as it relates the function directly to \( \cos(2\theta) \), whose periodicity can be calculated straightforwardly.

Properties of Sine Function

The sine function, denoted as \( \sin \theta \), is one of the fundamental trigonometric functions. It is known for its smooth oscillatory nature, repeating its pattern every full rotation of a circle, or mathematically every \(2\pi\).Some characteristics of the sine function include:

• Amplitude: The maximum value of \( \sin \theta \) is 1, and the minimum is -1.
• Periodicity: The standard period of \( \sin \theta \) is \(2\pi\).
• Even-Odd Nature: The function \( \sin \theta \) is odd, meaning \( \sin(-\theta) = -\sin \theta \).

However, when considering functions like \( \sin^2 \theta \), the periodicity changes due to transformations governed by trigonometric identities, such as the double angle formula. Understanding these concepts helps predict its behavior over intervals, aiding in calculations like determining its period.