If `sin theta + sin 5 theta = sin 3 theta and 0 lt theta lt ((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 to combine the left-hand side. The formula states that:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
In our case, let \( A = \theta \) and \( B = 5\theta \). Therefore, we have:
\[ \sin \theta + \sin 5\theta = 2 \sin\left(\frac{\theta + 5\theta}{2}\right) \cos\left(\frac{\theta - 5\theta}{2}\right) \]
This simplifies to:
\[ \sin \theta + \sin 5\theta = 2 \sin(3\theta) \cos(-2\theta) \]
Since \( \cos(-x) = \cos(x) \), we can write:
\[ \sin \theta + \sin 5\theta = 2 \sin(3\theta) \cos(2\theta) \]

### Step 2: Set the equation
Now we can equate this to the right-hand side of the original equation:
\[ 2 \sin(3\theta) \cos(2\theta) = \sin(3\theta) \]

### Step 3: Simplify the equation
Assuming \( \sin(3\theta) \neq 0 \) (which is valid since \( 0 < \theta < \frac{\pi}{2} \)), we can divide both sides by \( \sin(3\theta) \):
\[ 2 \cos(2\theta) = 1 \]

### Step 4: Solve for \( \cos(2\theta) \)
Now, we can solve for \( \cos(2\theta) \):
\[ \cos(2\theta) = \frac{1}{2} \]

### Step 5: Find the angle
The angle for which \( \cos(2\theta) = \frac{1}{2} \) is:
\[ 2\theta = 60^\circ \quad \text{or} \quad 2\theta = 300^\circ \]
However, since \( 0 < \theta < \frac{\pi}{2} \), we only consider:
\[ 2\theta = 60^\circ \]

### Step 6: Calculate \( \theta \)
Dividing by 2 gives:
\[ \theta = 30^\circ \]

### Final Answer
Thus, the value of \( \theta \) is:
\[ \theta = 30^\circ \]