For all `theta, sin theta + sin(theta + pi)+sin (2pi + theta)=`

Options

A. `-sin theta`
B. `sin theta`
C. `2 sin theta`
D. `3 sin theta`

Explanation

To solve the equation \( \sin \theta + \sin(\theta + \pi) + \sin(2\pi + \theta) \), we can follow these steps:

### Step 1: Understand the sine function properties
The sine function has a periodicity of \( 2\pi \). This means that:
- \( \sin(\theta + 2\pi) = \sin(\theta) \)
- \( \sin(\theta + \pi) = -\sin(\theta) \)

### Step 2: Substitute the values
Using the properties of the sine function, we can rewrite the terms in the equation:
- \( \sin(\theta + \pi) = -\sin(\theta) \)
- \( \sin(2\pi + \theta) = \sin(\theta) \)

### Step 3: Rewrite the original equation
Now, substituting these values back into the original equation:
\[ \sin \theta + \sin(\theta + \pi) + \sin(2\pi + \theta) = \sin \theta + (-\sin \theta) + \sin \theta \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ \sin \theta - \sin \theta + \sin \theta = 0 + \sin \theta = \sin \theta \]

### Final Result
Thus, we find that:
\[ \sin \theta + \sin(\theta + \pi) + \sin(2\pi + \theta) = \sin \theta \]

### Conclusion
The final answer is:
\[ \sin \theta \]

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