Question

If \(\sin 5x + \sin 3x + \sin x = 0\) then the value of x other than zero lying between \(0 \le x \le \frac{\pi}{2}\) is

• A. \(\frac{\pi}{6}\)
• B. \(\frac{\pi}{3}\)
• C. \(\frac{\pi}{12}\)
• D. \(\frac{\pi}{4}\)

Hint:

When solving equations with three or more sine or cosine terms, look for pairs of terms that can be combined using sum-to-product formulas to create a common factor. The choice of pairing is important. Pairing \(\sin 5x\) and \(\sin x\) was strategic because \((5x+x)/2 = 3x\), which matched the remaining term.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We need to solve the trigonometric equation sin 5x + sin 3x + sin x = 0 for a non-zero value of x in the interval [0, π/2].

Step 2: Key Formula or Approach (Alternate Method):
Group sin 5x and sin x using sum-to-product formula, then factor out common term sin 3x.

Step 3: Detailed Explanation:
Given: sin 5x + sin 3x + sin x = 0. Group first and last: (sin 5x + sin x) + sin 3x = 0. Sum-to-product: sin A + sin B = 2 sin((A+B)/2) cos((A-B)/2). sin 5x + sin x = 2 sin(3x) cos(2x). Equation becomes: 2 sin 3x cos 2x + sin 3x = 0. Factor: sin 3x (2 cos 2x + 1) = 0. Case 1: sin 3x = 0 → 3x = nπ → x = nπ/3. Non-zero in [0,π/2] gives x = π/3. Case 2: 2 cos 2x + 1 = 0 → cos 2x = -1/2 → 2x = 2π/3 → x = π/3. Both cases give same answer.

Step 4: Final Answer:
The non-zero value of x in the given interval that satisfies the equation is π/3.