If `sintheta-3sin2theta+sin3theta=costheta-3cos2theta+cos3theta`, then one of the general value is:

Options

A. `(npi)/(2)+pi/8`
B. `npi+pi/8`
C. `npi-pi/8`
D. `(npi)/(2)-pi/8`

Answer

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

### Step 1: Rearranging the Equation
We start with the given equation:
\[ \sin \theta - 3 \sin 2\theta + \sin 3\theta = \cos \theta - 3 \cos 2\theta + \cos 3\theta \]
We can rearrange this to group the sine terms on one side and the cosine terms on the other side:
\[ \sin 3\theta + \sin \theta - 3 \sin 2\theta = \cos 3\theta + \cos \theta - 3 \cos 2\theta \]

### Step 2: Applying Sum-to-Product Identities
Now, we can apply the sum-to-product identities. For sine, we have:
\[ \sin A + \sin B = 2 \sin\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
For cosine, we have:
\[ \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
Applying these identities:
- For \( \sin 3\theta + \sin \theta \):
\[ \sin 3\theta + \sin \theta = 2 \sin\left(2\theta\right) \cos\left(1\theta\right) \]
- For \( \cos 3\theta + \cos \theta \):
\[ \cos 3\theta + \cos \theta = 2 \cos\left(2\theta\right) \cos\left(1\theta\right) \]

### Step 3: Substitute Back into the Equation
Substituting these back into the equation gives:
\[ 2 \sin(2\theta) \cos(\theta) - 3 \sin 2\theta = 2 \cos(2\theta) \cos(\theta) - 3 \cos 2\theta \]

### Step 4: Factor Out Common Terms
We can factor out \( \sin 2\theta \) and \( \cos 2\theta \):
\[ \sin 2\theta (2 \cos \theta - 3) = \cos 2\theta (2 \cos \theta - 3) \]

### Step 5: Setting Up the Equation
This leads us to:
\[ \sin 2\theta = \cos 2\theta \]
This can be rewritten as:
\[ \tan 2\theta = 1 \]

### Step 6: Finding General Solutions
The equation \( \tan 2\theta = 1 \) has general solutions:
\[ 2\theta = n\pi + \frac{\pi}{4}, \quad n \in \mathbb{Z} \]
Thus, dividing by 2 gives:
\[ \theta = \frac{n\pi}{2} + \frac{\pi}{8} \]

### Final Answer
The general value of \( \theta \) is:
\[ \theta = \frac{n\pi}{2} + \frac{\pi}{8} \]