If `sin theta + sin 3 theta + sin 5theta = 0, 0 le theta le pi//2`, then `theta =`

Options

A. 0
B. `pi/4`
C. `pi/3`
D. `pi/2`

Answer

To solve the equation \( \sin \theta + \sin 3\theta + \sin 5\theta = 0 \) for \( 0 \leq \theta \leq \frac{\pi}{2} \), we can follow these steps:

### Step 1: Use the sine addition formula
We can group the terms \( \sin \theta + \sin 5\theta \) and use the sine addition formula:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Let \( A = 5\theta \) and \( B = \theta \):
\[ \sin \theta + \sin 5\theta = 2 \sin\left(\frac{5\theta + \theta}{2}\right) \cos\left(\frac{5\theta - \theta}{2}\right) = 2 \sin(3\theta) \cos(2\theta) \]

### Step 2: Rewrite the equation
Now we can rewrite the original equation:
\[ 2 \sin(3\theta) \cos(2\theta) + \sin(3\theta) = 0 \]
Factor out \( \sin(3\theta) \):
\[ \sin(3\theta)(2 \cos(2\theta) + 1) = 0 \]

### Step 3: Set each factor to zero
This gives us two equations to solve:
1. \( \sin(3\theta) = 0 \)
2. \( 2 \cos(2\theta) + 1 = 0 \)

### Step 4: Solve \( \sin(3\theta) = 0 \)
The sine function equals zero at integer multiples of \( \pi \):
\[ 3\theta = n\pi \quad \text{for } n = 0, 1, 2, \ldots \]
For \( n = 0 \):
\[ \theta = 0 \]
For \( n = 1 \):
\[ \theta = \frac{\pi}{3} \]
For \( n = 2 \):
\[ \theta = \frac{2\pi}{3} \quad (\text{not in the interval } [0, \frac{\pi}{2}]) \]

### Step 5: Solve \( 2 \cos(2\theta) + 1 = 0 \)
Rearranging gives:
\[ \cos(2\theta) = -\frac{1}{2} \]
The cosine function equals \(-\frac{1}{2}\) at:
\[ 2\theta = \frac{2\pi}{3} + 2k\pi \quad \text{or} \quad 2\theta = \frac{4\pi}{3} + 2k\pi \]
For \( k = 0 \):
\[ 2\theta = \frac{2\pi}{3} \implies \theta = \frac{\pi}{3} \]
For \( k = 0 \):
\[ 2\theta = \frac{4\pi}{3} \implies \theta = \frac{2\pi}{3} \quad (\text{not in the interval } [0, \frac{\pi}{2}]) \]

### Step 6: Final solutions
The possible values of \( \theta \) are \( 0 \) and \( \frac{\pi}{3} \). Since \( 0 \) is included in the interval and \( \frac{\pi}{3} \) is also included, we conclude:
\[ \theta = 0 \quad \text{or} \quad \theta = \frac{\pi}{3} \]