What is the value of sin `(-1125^(@))`

Answer

To find the value of \( \sin(-1125^\circ) \), we can follow these steps:

### Step 1: Use the property of sine for negative angles
We know that:
\[ \sin(-\theta) = -\sin(\theta) \]
Thus, we can rewrite:
\[ \sin(-1125^\circ) = -\sin(1125^\circ) \]

### Step 2: Simplify the angle
The angle \( 1125^\circ \) is greater than \( 360^\circ \). To find an equivalent angle within the range of \( 0^\circ \) to \( 360^\circ \), we can subtract multiples of \( 360^\circ \):
\[ 1125^\circ - 360^\circ \times 3 = 1125^\circ - 1080^\circ = 45^\circ \]
So, we have:
\[ \sin(1125^\circ) = \sin(45^\circ) \]

### Step 3: Find the sine of \( 45^\circ \)
We know that:
\[ \sin(45^\circ) = \frac{1}{\sqrt{2}} \]

### Step 4: Substitute back to find \( \sin(-1125^\circ) \)
Now substituting back, we have:
\[ \sin(-1125^\circ) = -\sin(1125^\circ) = -\sin(45^\circ) = -\frac{1}{\sqrt{2}} \]

### Step 5: Rationalize the denominator
To express \( -\frac{1}{\sqrt{2}} \) in a more standard form, we can rationalize the denominator:
\[ -\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = -\frac{\sqrt{2}}{2} \]

### Final Answer
Thus, the value of \( \sin(-1125^\circ) \) is:
\[ \sin(-1125^\circ) = -\frac{\sqrt{2}}{2} \]

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