What is the value of sin `(1110^@)`?

Options

A. 1
B. `1/2`
C. `(1)/(sqrt2)`
D. `(sqrt3)/(2)`

Answer

To find the value of \( \sin(1110^\circ) \), we can use the periodic property of the sine function. The sine function has a period of \( 360^\circ \), which means that:

\[ \sin(\theta) = \sin(\theta + 360^\circ n) \]

where \( n \) is any integer.

### Step-by-step solution:

1. **Reduce the angle**: Since \( 1110^\circ \) is greater than \( 360^\circ \), we can reduce it by subtracting multiples of \( 360^\circ \) until we find an equivalent angle within the range of \( 0^\circ \) to \( 360^\circ \).

\[ 1110^\circ - 360^\circ \times 3 = 1110^\circ - 1080^\circ = 30^\circ \]

So, \( \sin(1110^\circ) = \sin(30^\circ) \).

2. **Find the sine value**: Now we need to find \( \sin(30^\circ) \). From trigonometric values, we know:

\[ \sin(30^\circ) = \frac{1}{2} \]

3. **Conclusion**: Therefore, the value of \( \sin(1110^\circ) \) is:

\[ \sin(1110^\circ) = \frac{1}{2} \]

### Final Answer:
The value of \( \sin(1110^\circ) \) is \( \frac{1}{2} \).

---