Question

Number of solutions of the equation \[ \sin\theta+\sin3\theta+\sin5\theta=0 \] in \([-\pi,\pi]\) is

• A. 5
• B. 7
• C. 9
• D. 11 \bigskip

Hint:

Always try pairing sine terms to reduce expression order.

Answer 1

The Correct Option is C

Concept: Use sum-to-product identities to reduce trigonometric sums.

Step 1: Group terms.
\[ \sin\theta+\sin5\theta=2\sin3\theta\cos2\theta \] So equation becomes: \[ 2\sin3\theta\cos2\theta+\sin3\theta=0 \]

Step 2: Factorize.
\[ \sin3\theta(2\cos2\theta+1)=0 \]

Step 3: Solve cases.
Case 1: \[ \sin3\theta=0 \Rightarrow 3\theta=n\pi \Rightarrow \theta=\frac{n\pi}{3} \] In \([-\pi,\pi]\), \(n=-3,-2,\dots,3\Rightarrow 7\) solutions. Case 2: \[ 2\cos2\theta+1=0 \Rightarrow \cos2\theta=-\frac12 \] \[ 2\theta=\frac{2\pi}{3},\frac{4\pi}{3} \Rightarrow \theta=\frac{\pi}{3},\frac{2\pi}{3},\dots \] Gives 2 more solutions.

Step 4: Total solutions.
\[ 7+2=9 \] \[ \boxed{(C)} \]