If we convert `sin(-566^(@))` to same trigonometrical ratio of a positive angle lying between `0^(@) and 45^(@)`, then we get :

Options

A. `cos26^(@)`
B. `-cos26^(@)`
C. `sin26^(@)`
D. `-sin26^(@)`

Answer

To solve the problem of converting `sin(-566°)` to the same trigonometric ratio of a positive angle lying between `0°` and `45°`, we will follow these steps:

### Step-by-Step Solution:

1. **Use the property of sine for negative angles**:
\[ \sin(-\theta) = -\sin(\theta) \]
Applying this property:
\[ \sin(-566°) = -\sin(566°) \]

**Hint**: Remember that the sine function is odd, which means that the sine of a negative angle is the negative of the sine of the corresponding positive angle.

2. **Reduce the angle 566° to a standard position**:
To find an equivalent angle, we can subtract multiples of 360° (since sine has a periodicity of 360°):
\[ 566° - 360° = 206° \]
So, we have:
\[ \sin(566°) = \sin(206°) \]
Therefore:
\[ \sin(-566°) = -\sin(206°) \]

**Hint**: When reducing angles, always subtract 360° until the angle is between 0° and 360°.

3. **Determine the reference angle for 206°**:
The angle 206° is in the third quadrant. To find the reference angle, we subtract 180°:
\[ 206° - 180° = 26° \]
Thus:
\[ \sin(206°) = -\sin(26°) \]
Therefore:
\[ \sin(-566°) = -(-\sin(26°)) = \sin(26°) \]

**Hint**: The reference angle is found by subtracting the angle from 180° or 360° depending on the quadrant.

4. **Final answer**:
Since we need the sine of an angle between `0°` and `45°`, we have:
\[ \sin(-566°) = \sin(26°) \]

Thus, the final answer is:
\[ \sin(-566°) = \sin(26°) \]

### Conclusion:
The sine of `-566°` converts to the sine of `26°`, which lies between `0°` and `45°`.