Find all the values of ` theta ` satisfying the equation ` sin theta + sin 5 theta = sin 3 theta,` such that `0 le theta le pi.`

Answer

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

### Step 1: Use the sum-to-product identities
We know that:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
In our case, let \( A = 5\theta \) and \( B = \theta \). Thus, we can rewrite the left-hand side:
\[ \sin \theta + \sin 5\theta = 2 \sin\left(\frac{5\theta + \theta}{2}\right) \cos\left(\frac{5\theta - \theta}{2}\right) = 2 \sin(3\theta) \cos(2\theta) \]

### Step 2: Set the equation
Now, we can set the equation:
\[ 2 \sin(3\theta) \cos(2\theta) = \sin(3\theta) \]

### Step 3: Factor the equation
Rearranging gives:
\[ 2 \sin(3\theta) \cos(2\theta) - \sin(3\theta) = 0 \]
Factoring out \( \sin(3\theta) \):
\[ \sin(3\theta)(2 \cos(2\theta) - 1) = 0 \]

### Step 4: Solve the factors
This gives us two cases to solve:
1. \( \sin(3\theta) = 0 \)
2. \( 2 \cos(2\theta) - 1 = 0 \)

#### Case 1: \( \sin(3\theta) = 0 \)
The general solution for \( \sin x = 0 \) is:
\[ 3\theta = n\pi \quad \Rightarrow \quad \theta = \frac{n\pi}{3} \]
For \( 0 \leq \theta \leq \pi \):
- If \( n = 0 \): \( \theta = 0 \)
- If \( n = 1 \): \( \theta = \frac{\pi}{3} \)
- If \( n = 2 \): \( \theta = \frac{2\pi}{3} \)
- If \( n = 3 \): \( \theta = \pi \)

#### Case 2: \( 2 \cos(2\theta) - 1 = 0 \)
Solving gives:
\[ \cos(2\theta) = \frac{1}{2} \]
The general solution for \( \cos x = \frac{1}{2} \) is:
\[ 2\theta = 2n\pi \pm \frac{\pi}{3} \quad \Rightarrow \quad \theta = n\pi \pm \frac{\pi}{6} \]
For \( 0 \leq \theta \leq \pi \):
- If \( n = 0 \): \( \theta = \frac{\pi}{6} \) and \( \theta = -\frac{\pi}{6} \) (not valid)
- If \( n = 1 \): \( \theta = \pi + \frac{\pi}{6} \) (not valid)
- If \( n = 1 \): \( \theta = \pi - \frac{\pi}{6} = \frac{5\pi}{6} \)

### Step 5: Compile all solutions
The valid solutions for \( \theta \) in the range \( 0 \leq \theta \leq \pi \) are:
- \( \theta = 0 \)
- \( \theta = \frac{\pi}{3} \)
- \( \theta = \frac{2\pi}{3} \)
- \( \theta = \pi \)
- \( \theta = \frac{\pi}{6} \)
- \( \theta = \frac{5\pi}{6} \)

### Final Answer
Thus, the complete set of solutions is:
\[ \theta = 0, \frac{\pi}{6}, \frac{\pi}{3}, \frac{2\pi}{3}, \frac{5\pi}{6}, \pi \]