If `sintheta+sin5theta=sin3thetaand0lttheta((pi)/(2))` , what is the value of `theta` (in degrees) ?

Options

A. 30
B. 45
C. 60
D. 75

Answer

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

### Step 1: Use the Sine Addition Formula
We can use the sine addition formula:
\[ \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]
In our case, let \( A = \theta \) and \( B = 5\theta \).

### Step 2: Apply the Formula
Applying the formula:
\[ \sin \theta + \sin 5\theta = 2 \sin\left(\frac{\theta + 5\theta}{2}\right) \cos\left(\frac{\theta - 5\theta}{2}\right) \]
This simplifies to:
\[ \sin \theta + \sin 5\theta = 2 \sin(3\theta) \cos(2\theta) \]

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

### Step 4: Rearrange the Equation
Rearranging gives us:
\[ 2 \sin(3\theta) \cos(2\theta) - \sin(3\theta) = 0 \]
Factoring out \( \sin(3\theta) \):
\[ \sin(3\theta)(2 \cos(2\theta) - 1) = 0 \]

### Step 5: Solve the Factors
This gives us two cases to solve:
1. \( \sin(3\theta) = 0 \)
2. \( 2 \cos(2\theta) - 1 = 0 \)

### Step 6: Solve \( \sin(3\theta) = 0 \)
For \( \sin(3\theta) = 0 \):
\[ 3\theta = n\pi \quad (n \in \mathbb{Z}) \]
Thus:
\[ \theta = \frac{n\pi}{3} \]
Considering the interval \( 0 < \theta < \frac{\pi}{2} \), we take \( n = 1 \):
\[ \theta = \frac{\pi}{3} = 60^\circ \]

### Step 7: Solve \( 2 \cos(2\theta) - 1 = 0 \)
For \( 2 \cos(2\theta) - 1 = 0 \):
\[ \cos(2\theta) = \frac{1}{2} \]
This gives:
\[ 2\theta = 60^\circ + k \cdot 360^\circ \quad \text{or} \quad 2\theta = 300^\circ + k \cdot 360^\circ \]
Thus:
\[ \theta = 30^\circ + k \cdot 180^\circ \quad \text{or} \quad \theta = 150^\circ + k \cdot 180^\circ \]
Considering \( k = 0 \):
\[ \theta = 30^\circ \quad \text{and} \quad \theta = 75^\circ \text{ (not in the range)} \]

### Conclusion
The possible values for \( \theta \) are \( 60^\circ \) and \( 30^\circ \). However, since we are looking for values in the range \( 0 < \theta < \frac{\pi}{2} \), the valid solution is:
\[ \theta = 30^\circ \]