If `theta=4530^(@)`, then `"sin" 4530^(@)` is :

Options

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

Answer

To find the value of \(\sin 4530^\circ\), we can follow these steps:

### Step 1: Reduce the angle
First, we need to reduce \(4530^\circ\) to an equivalent angle within the range of \(0^\circ\) to \(360^\circ\). We can do this by subtracting multiples of \(360^\circ\) from \(4530^\circ\).

\[ 4530^\circ \div 360^\circ = 12.5833 \]

This means \(4530^\circ\) is equivalent to \(12\) full rotations of \(360^\circ\) plus a remainder. To find the remainder, we calculate:

\[ 12 \times 360^\circ = 4320^\circ \]
\[ 4530^\circ - 4320^\circ = 210^\circ \]

So, \(4530^\circ\) is equivalent to \(210^\circ\).

### Step 2: Find \(\sin 210^\circ\)
Next, we need to find \(\sin 210^\circ\). We know that \(210^\circ\) is in the third quadrant, where sine values are negative.

\[ \sin 210^\circ = \sin(180^\circ + 30^\circ) = -\sin 30^\circ \]

### Step 3: Calculate \(\sin 30^\circ\)
We know that:

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

### Step 4: Apply the sign from the quadrant
Since \(210^\circ\) is in the third quadrant, we have:

\[ \sin 210^\circ = -\sin 30^\circ = -\frac{1}{2} \]

### Final Answer
Thus, the value of \(\sin 4530^\circ\) is:

\[ \sin 4530^\circ = -\frac{1}{2} \]

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