Problem 40 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Solution of the equation \(\sin ^{3} \theta+\sin \theta \cos \theta+\cos ^{3} \theta=1\) is (A) \(\theta=2 n \pi+\frac{\pi}{4}\) (B) \(\theta=2 n \pi-\frac{\pi}{4}\) (C) \(\theta=2 n \pi+\frac{\pi}{2}\) (D) \(\theta=2 n \pi\)

Step-by-Step Solution

Answer

The solution is \(\theta = 2n\pi\) (D).

Step 1: Analyze the Equation

The equation given is \( \sin^3 \theta + \sin \theta \cos \theta + \cos^3 \theta = 1 \). We need to explore solutions where trigonometric functions sum to a value of 1.

Step 2: Simplify Using Trigonometric Identities

Recall the identity \( \sin^2 \theta + \cos^2 \theta = 1 \). Here the powers and multiplies complicate direct application, therefore we focus on possible simplifications such as \( \sin \theta \cos \theta \) which can be rewritten in terms of product identities like \( 2 \sin \theta \cos \theta = \sin 2\theta \).

Step 3: Guess and Check for Specific Angles

Check common angles. For example, substituting \( \theta = 0 \), yields \( \sin 0 = 0 \) and \( \cos 0 = 1 \), leading to \( 0^3 + 0 \times 1 + 1^3 = 1 \). Therefore, \( \theta = 0 \) (or multiples of \(2\pi\)) is a solution.

Step 4: Confirm the General Solution

The pattern \( \theta = 2n\pi \) fits with our earlier evaluation at specific angles and reflects a valid periodic behavior seen in trigonometric functions.

Key Concepts

Trigonometric Identities

In trigonometry, identities are essential tools that simplify complex equations and help in solving trigonometric problems effectively. An identity is an equality that holds true for any value of the variable involved. One of the most fundamental trigonometric identities is

• \( \sin^2 \theta + \cos^2 \theta = 1\)

This identity is derived from the Pythagorean theorem and expresses a relationship between the sine and cosine functions of an angle \(\theta\). Trigonometric identities are often used to simplify product terms; for example, the identity for the product \(\sin \theta \cos \theta\) is given by:

• \(2 \sin \theta \cos \theta = \sin 2\theta\)

Applying these identities can simplify complex expressions and make it easier to identify solutions. Identifying these identities is critically beneficial, especially when dealing with cubic or higher powers in trigonometric equations. So, whenever you find these high-degree terms, look for ways to simplify using these identities.

Angle Solutions

Solving trigonometric equations often revolves around finding specific angles that satisfy the equation. In the given problem, the equation \(\sin^3 \theta + \sin \theta \cos \theta + \cos^3 \theta = 1\) is simplified by testing common angles, such as \(0, \frac{\pi}{4}, \frac{\pi}{2}\), among others. To solve these types of equations:

• Substitute angles that result in simple values for \(\sin\) and \(\cos\), such as \(\frac{\pi}{4}\) where \(\sin \theta = \cos \theta = \frac{\sqrt{2}}{2}\).
• Recognize that if an angle like \(0\) satisfies the equation, the solution set can include all its periodic repetitions, which are multiples of \(2\pi\).

Through this method of angle substitution and analysis, it was found that the general solution pattern \(\theta = 2n\pi\) appears, depicting the periodic nature of trigonometric functions, which repeat their values in cycles, or periods. Therefore, multiple solution strategies can lead you to the correct general angular solution.

Sin and Cos Functions

The sine and cosine functions are fundamental in trigonometry, characterized by their periodic and oscillatory nature. Understanding these functions is key to solving equations and finding their angular solutions.Key properties of \(\sin\) and \(\cos\):

• The sine function, \(\sin \theta\), starts at zero when \(\theta\) is zero and oscillates between -1 and 1 with a period of \(2\pi\).
• The cosine function, \(\cos \theta\), starts at one when \(\theta\) is zero and also oscillates with the same amplitude and period as sine.

When combined in equations, the relationships and identities formed between these functions allow for solving even complex equations. Their graphical representation helps visualize their behavior over a period and recognize their repeating nature.For example, in the problem presented, recognizing how \(\sin\) and \(\cos\) values vary at standard angles provides a simplified approach to confirming solutions like \(\theta = 2n\pi\). These insights are invaluable for deconstructing and solving trigonometric equations efficiently.