Question

Let \[ S=\{x\in[-\pi,\pi]:\sin x(\sin x+\cos x)=a,\; a\in\mathbb{Z}\}. \] Then \(n(S)\) is equal to:

• A. \(3\)
• B. \(6\)
• C. \(7\)
• D. \(9\)

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We need to find the number of solutions for the equation \(\sin^2 x + \sin x \cos x = a\) where \(a\) is an integer and \(x \in [-\pi, \pi]\). We first determine the range of the left-hand side function to find which integers \(a\) are possible.
Step 2: Key Formula or Approach:
1. Simplify function: \(f(x) = \sin^2 x + \sin x \cos x = \frac{1 - \cos 2x}{2} + \frac{\sin 2x}{2}\).
2. Transformation: \(A \sin \theta + B \cos \theta = \sqrt{A^2+B^2} \sin(\theta + \phi)\).
Step 3: Detailed Explanation:
Let \(f(x) = \frac{1}{2} + \frac{1}{2}(\sin 2x - \cos 2x)\).
Using the transformation, \(\sin 2x - \cos 2x = \sqrt{2} \sin(2x - \pi/4)\).
So, \(f(x) = \frac{1}{2} + \frac{1}{\sqrt{2}} \sin(2x - \frac{\pi}{4})\).
Range of \(\sin(2x - \frac{\pi}{4})\) is \([-1, 1]\).
Range of \(f(x)\) is \([\frac{1}{2} - \frac{1}{\sqrt{2}}, \frac{1}{2} + \frac{1}{\sqrt{2}}]\).
Approximate values: \(\frac{1}{\sqrt{2}} \approx 0.707\).
Range \(\approx [0.5 - 0.707, 0.5 + 0.707] \approx [-0.207, 1.207]\).
The only integers in this range are \(a = 0\) and \(a = 1\).
Case 1: \(a = 0\)
\(\sin x (\sin x + \cos x) = 0 \implies \sin x = 0\) or \(\tan x = -1\).
For \(x \in [-\pi, \pi]\): \(\sin x = 0 \implies x = \{-\pi, 0, \pi\}\) (3 solutions).
\(\tan x = -1 \implies x = \{-\pi/4, 3\pi/4\}\) (2 solutions).
Total solutions for \(a=0\) is \(3 + 2 = 5\).
Case 2: \(a = 1\)
\(\frac{1}{2} + \frac{1}{\sqrt{2}} \sin(2x - \frac{\pi}{4}) = 1 \implies \sin(2x - \frac{\pi}{4}) = \frac{1}{\sqrt{2}}\).
Let \(\theta = 2x - \frac{\pi}{4}\). Since \(x \in [-\pi, \pi]\), \(\theta \in [-2\pi - \frac{\pi}{4}, 2\pi - \frac{\pi}{4}] = [-\frac{9\pi}{4}, \frac{7\pi}{4}]\).
Solutions for \(\sin \theta = \frac{1}{\sqrt{2}}\) in this range:
\(\theta = \{ \dots, -2\pi+\frac{\pi}{4}, -2\pi+\frac{3\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4} \} = \{-\frac{7\pi}{4}, -\frac{5\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4}\}\).
All these values are within the range. This gives 4 solutions.
Total elements in set \(S = 5 + 4 = 9\).
Step 4: Final Answer:
The number of elements \(n(S)\) is 9.