Without using tables, give the value of the following :
`sin4530^(@)`

Answer

To find the value of \( \sin 4530^\circ \) without using tables, we can follow these steps:

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

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

This means \( 4530^\circ \) is equivalent to \( 12 \) complete rotations (since \( 12 \times 360 = 4320 \)) plus some extra degrees.

Calculating the extra degrees:

\[ 4530^\circ - 4320^\circ = 210^\circ \]

So,

\[ \sin 4530^\circ = \sin 210^\circ \]

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

The reference angle for \( 210^\circ \) is:

\[ 210^\circ - 180^\circ = 30^\circ \]

Thus,

\[ \sin 210^\circ = -\sin 30^\circ \]

### Step 3: Use known sine values
We know that:

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

Therefore,

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

### Conclusion
Finally, we conclude that:

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

### Final Answer:
\[ \sin 4530^\circ = -\frac{1}{2} \]

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