If `sin 7 theta + sin 4 theta + sin theta = 0, 0 le theta le pi //2` then ` theta`=

Options

A. 0
B. `40^@`
C. `45^@`
D. `80^@`

Answer

To solve the equation \( \sin 7\theta + \sin 4\theta + \sin \theta = 0 \) for \( 0 \leq \theta \leq \frac{\pi}{2} \), we can follow these steps:

### Step 1: Use the Sum-to-Product Identities
We can use the sum-to-product identities for sine, which states:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
We will apply this identity to \( \sin 7\theta + \sin \theta \).

Let \( A = 7\theta \) and \( B = \theta \):
\[ \sin 7\theta + \sin \theta = 2 \sin\left(\frac{7\theta + \theta}{2}\right) \cos\left(\frac{7\theta - \theta}{2}\right) = 2 \sin(4\theta) \cos(3\theta) \]

### Step 2: Substitute Back into the Equation
Now we can substitute this back into the original equation:
\[ 2 \sin(4\theta) \cos(3\theta) + \sin 4\theta = 0 \]
Factoring out \( \sin 4\theta \):
\[ \sin 4\theta (2 \cos 3\theta + 1) = 0 \]

### Step 3: Set Each Factor to Zero
Now we have two factors to consider:
1. \( \sin 4\theta = 0 \)
2. \( 2 \cos 3\theta + 1 = 0 \)

#### For \( \sin 4\theta = 0 \):
This gives us:
\[ 4\theta = n\pi \quad \text{for } n = 0, 1, 2, \ldots \]
Thus:
\[ \theta = \frac{n\pi}{4} \]
Considering \( 0 \leq \theta \leq \frac{\pi}{2} \):
- For \( n = 0 \): \( \theta = 0 \)
- For \( n = 1 \): \( \theta = \frac{\pi}{4} \)
- For \( n = 2 \): \( \theta = \frac{\pi}{2} \)

#### For \( 2 \cos 3\theta + 1 = 0 \):
This simplifies to:
\[ \cos 3\theta = -\frac{1}{2} \]
The angles where cosine equals \(-\frac{1}{2}\) are:
\[ 3\theta = \frac{2\pi}{3} + 2k\pi \quad \text{and} \quad 3\theta = \frac{4\pi}{3} + 2k\pi \quad \text{for } k \in \mathbb{Z} \]
Thus:
\[ \theta = \frac{2\pi}{9} + \frac{2k\pi}{3} \quad \text{and} \quad \theta = \frac{4\pi}{9} + \frac{2k\pi}{3} \]
Considering \( 0 \leq \theta \leq \frac{\pi}{2} \):
- For \( k = 0 \):
- \( \theta = \frac{2\pi}{9} \) (valid)
- \( \theta = \frac{4\pi}{9} \) (valid)
- For \( k = 1 \):
- \( \theta = \frac{2\pi}{9} + \frac{2\pi}{3} \) (not valid)
- \( \theta = \frac{4\pi}{9} + \frac{2\pi}{3} \) (not valid)

### Step 4: Collect All Valid Solutions
The valid solutions for \( \theta \) are:
- \( \theta = 0 \)
- \( \theta = \frac{\pi}{4} \)
- \( \theta = \frac{2\pi}{9} \)
- \( \theta = \frac{4\pi}{9} \)

### Step 5: Convert to Degrees (if necessary)
- \( 0 \) degrees
- \( 45 \) degrees
- \( \frac{2\pi}{9} \approx 40 \) degrees
- \( \frac{4\pi}{9} \approx 80 \) degrees

### Final Answer
Thus, the values of \( \theta \) that satisfy the equation in the given range are:
\[ \theta = 0, 40^\circ, 45^\circ, 80^\circ \]