Question

If $x \in (-\pi,\pi)$ then the number of solutions of the equation $2 \sin x \sin 3x \sin 5x + \sin 5x \cos 4x = 0$ is

• A. 14
• B. 12
• C. 13
• D. 9

Hint:

When solving trigonometric equations, first try to factor the expression. This breaks the problem into simpler, separate cases. Always check for common solutions between the cases before stating the final count.

Answer 1

The Correct Option is C

The given equation is $2 \sin x \sin 3x \sin 5x + \sin 5x \cos 4x = 0$.
Factor out the common term $\sin 5x$:
$\sin 5x (2 \sin x \sin 3x + \cos 4x) = 0$.
This gives two possibilities for the solutions:
Case 1: $\sin 5x = 0$.
This implies $5x = n\pi$ for any integer $n$. So, $x = \frac{n\pi}{5}$.
We are given the interval $x \in (-\pi, \pi)$, so $-\pi<\frac{n\pi}{5}<\pi$.
Dividing by $\pi$ gives $-1<\frac{n}{5}<1$, which means $-5<n<5$.
The possible integer values for $n$ are $-4, -3, -2, -1, 0, 1, 2, 3, 4$. This gives 9 distinct solutions.
Case 2: $2 \sin x \sin 3x + \cos 4x = 0$.
Using the product-to-sum formula $2 \sin A \sin B = \cos(A-B) - \cos(A+B)$, we have:
$2 \sin 3x \sin x = \cos(3x-x) - \cos(3x+x) = \cos(2x) - \cos(4x)$.
Substituting this into the equation for Case 2:
$(\cos 2x - \cos 4x) + \cos 4x = 0$.
$\cos 2x = 0$.
This implies $2x = (2k+1)\frac{\pi}{2}$ for any integer $k$. So, $x = (2k+1)\frac{\pi}{4}$.
For the interval $x \in (-\pi, \pi)$, we have $-\pi<(2k+1)\frac{\pi}{4}<\pi$.
Multiplying by $4/\pi$ gives $-4<2k+1<4$, which means $-5<2k<3$, or $-2.5<k<1.5$.
The possible integer values for $k$ are $-2, -1, 0, 1$. This gives 4 distinct solutions: $-\frac{3\pi}{4}, -\frac{\pi}{4}, \frac{\pi}{4}, \frac{3\pi}{4}$.
We must check if there is any overlap between the solutions from Case 1 and Case 2. An overlap would occur if $\frac{n}{5} = \frac{2k+1}{4}$ for some allowed integers $n, k$. This gives $4n = 5(2k+1)$. The left side is even, while the right side is odd. This is impossible, so there are no common solutions.
Total number of solutions = (Solutions from Case 1) + (Solutions from Case 2) = $9 + 4 = 13$.