The value of `sin10^@+sin20^@+sin30^@...+sin360^@` is equal to -

Options

A. 1
B. 0
C. `-1`
D. `1//2`

Answer

To find the value of \( \sin 10^\circ + \sin 20^\circ + \sin 30^\circ + \ldots + \sin 360^\circ \), we can follow these steps:

### Step 1: Identify the range of angles
We are summing the sine values from \( \sin 10^\circ \) to \( \sin 360^\circ \). The angles are \( 10^\circ, 20^\circ, 30^\circ, \ldots, 360^\circ \).

### Step 2: Group the sine values
Notice that sine has a periodic property and symmetry. Specifically, we can pair the sine values from \( 10^\circ \) to \( 170^\circ \) with those from \( 190^\circ \) to \( 350^\circ \):
- \( \sin 190^\circ = \sin(180^\circ + 10^\circ) = -\sin 10^\circ \)
- \( \sin 200^\circ = \sin(180^\circ + 20^\circ) = -\sin 20^\circ \)
- \( \sin 210^\circ = \sin(180^\circ + 30^\circ) = -\sin 30^\circ \)
- ...
- \( \sin 350^\circ = \sin(360^\circ - 10^\circ) = -\sin 10^\circ \)

### Step 3: Write the complete sum
The complete sum can be expressed as:
\[ (\sin 10^\circ + \sin 20^\circ + \sin 30^\circ + \ldots + \sin 170^\circ) + (\sin 180^\circ) + (\sin 190^\circ + \sin 200^\circ + \sin 210^\circ + \ldots + \sin 360^\circ) \]

### Step 4: Simplify the sum
The sine values from \( 190^\circ \) to \( 350^\circ \) can be rewritten using the identities:
\[ \sin 190^\circ + \sin 200^\circ + \sin 210^\circ + \ldots + \sin 360^\circ = -(\sin 10^\circ + \sin 20^\circ + \sin 30^\circ + \ldots + \sin 170^\circ) \]

### Step 5: Combine the terms
Thus, we can combine the terms:
\[ \sin 10^\circ + \sin 20^\circ + \ldots + \sin 170^\circ + \sin 180^\circ - (\sin 10^\circ + \sin 20^\circ + \ldots + \sin 170^\circ) = \sin 180^\circ \]

### Step 6: Evaluate the sine values
Since \( \sin 180^\circ = 0 \), we have:
\[ \sin 10^\circ + \sin 20^\circ + \ldots + \sin 360^\circ = 0 \]

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

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