The value of `sin10^(@)+2sin20^(@)+sin30^(@)+….+sin360^(@)` is equal to

Options

A. 0
B. 1
C. `sqrt(3)`
D. 2

Answer

To solve the problem \( \sin 10^\circ + 2\sin 20^\circ + \sin 30^\circ + \ldots + \sin 360^\circ \), we can break it down step by step.

### Step 1: Write down the series
The series can be expressed as:
\[ S = \sin 10^\circ + \sin 20^\circ + \sin 30^\circ + \ldots + \sin 360^\circ \]

### Step 2: Identify the symmetry in sine function
The sine function has a property where:
\[ \sin(360^\circ - x) = -\sin x \]
This means that for every angle \( x \) in the series, there is a corresponding angle \( 360^\circ - x \) that will cancel it out.

### Step 3: Pair the terms
We can pair the terms in the series:
- \( \sin 10^\circ \) and \( \sin 350^\circ \)
- \( \sin 20^\circ \) and \( \sin 340^\circ \)
- \( \sin 30^\circ \) and \( \sin 330^\circ \)
- ...
- \( \sin 170^\circ \) and \( \sin 190^\circ \)
- \( \sin 180^\circ \) stands alone.

Each pair sums to zero:
\[ \sin x + \sin(360^\circ - x) = \sin x - \sin x = 0 \]

### Step 4: Calculate the total
Since all pairs cancel out, we are left with:
\[ S = 0 \]

### Step 5: Include the extra term
The only term that does not have a pair is \( \sin 180^\circ \), which is also equal to 0:
\[ \sin 180^\circ = 0 \]

### Conclusion
Thus, the total sum \( S \) is:
\[ S = 0 + 0 = 0 \]

### Final Answer
The value of \( \sin 10^\circ + 2\sin 20^\circ + \sin 30^\circ + \ldots + \sin 360^\circ \) is equal to \( 0 \).

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