Smallest positive value of `theta` satisfying `8sinthetacos2thetasin3thetacos4theta=cos6theta` is

Options

A. `(pi)/(18)`
B. `(pi)/(22)`
C. `(pi)/(24)`
D. none of these

Answer

To solve the equation \( 8 \sin \theta \cos 2\theta \sin 3\theta \cos 4\theta = \cos 6\theta \), we will follow these steps:

### Step 1: Multiply both sides by \( \cos \theta \)
We start by multiplying both sides of the equation by \( \cos \theta \):
\[ \cos \theta (8 \sin \theta \cos 2\theta \sin 3\theta \cos 4\theta) = \cos 6\theta \cos \theta \]

### Step 2: Use the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \)
Using the identity \( \sin 2\theta = 2 \sin \theta \cos \theta \), we can rewrite the equation:
\[ 8 \sin \theta \cos 2\theta \sin 3\theta \cos 4\theta = \cos 6\theta \cos \theta \]
This can be simplified to:
\[ 4 \sin 2\theta \cos 2\theta \sin 3\theta \cos 4\theta = \cos 6\theta \cos \theta \]

### Step 3: Use the identity \( \sin 4\theta = 2 \sin 2\theta \cos 2\theta \)
Next, we can apply the identity again:
\[ 4 \sin 4\theta \sin 3\theta \cos 4\theta = \cos 6\theta \cos \theta \]

### Step 4: Use the identity \( \sin 8\theta = 2 \sin 4\theta \cos 4\theta \)
We can express the left-hand side using the identity for \( \sin 8\theta \):
\[ 2 \sin 8\theta \sin 3\theta = \cos 6\theta \cos \theta \]

### Step 5: Use the sum-to-product identities
Now, we can use the sum-to-product identities:
\[ \cos 6\theta \cos \theta = \frac{1}{2} (\cos(6\theta + \theta) + \cos(6\theta - \theta)) \]
This leads us to:
\[ \cos 6\theta \cos \theta = \frac{1}{2} (\cos 7\theta + \cos 5\theta) \]

### Step 6: Set up the equation
Now we have:
\[ 2 \sin 8\theta \sin 3\theta = \frac{1}{2} (\cos 7\theta + \cos 5\theta) \]

### Step 7: Solve for \( \theta \)
From here, we can simplify and solve for \( \theta \). We will set:
\[ \cos 11\theta + \cos 7\theta = 0 \]
This leads to:
\[ \cos 9\theta = 0 \quad \text{or} \quad \cos 2\theta = 0 \]

### Step 8: Find the values of \( \theta \)
1. For \( \cos 9\theta = 0 \):
\[ 9\theta = \frac{\pi}{2} + n\pi \implies \theta = \frac{\pi}{18} + \frac{n\pi}{9} \]
2. For \( \cos 2\theta = 0 \):
\[ 2\theta = \frac{\pi}{2} + m\pi \implies \theta = \frac{\pi}{4} + \frac{m\pi}{2} \]

### Step 9: Determine the smallest positive value of \( \theta \)
The smallest positive value from the solutions is:
\[ \theta = \frac{\pi}{18} \]

### Conclusion
Thus, the smallest positive value of \( \theta \) satisfying the equation is:
\[ \theta = \frac{\pi}{18} \]

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