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The problem asks for the coordinates of a point \( P \) that lies on the unit circle with the center at \( O = (0, 0) \). The line from the positive \( x \)-axis to the point \( P \) forms an angle of \( \alpha = 251^\circ \).

### Step-by-step solution:
1. **Understanding the unit circle**:
   - The equation of the unit circle is \( x^2 + y^2 = 1 \), where any point \( P(x, y) \) on the unit circle can be represented using trigonometric functions based on the angle \( \alpha \).
   - For an angle \( \alpha \) measured counterclockwise from the positive \( x \)-axis, the coordinates of point \( P \) are:
     \[
     P(x, y) = (\cos \alpha, \sin \alpha)
     \]
   
2. **Given information**:
   - The angle \( \alpha = 251^\circ \).
   - We need to calculate \( \cos(251^\circ) \) and \( \sin(251^\circ) \) to find the coordinates of \( P \), and then round the result to two decimal places.

3. **Calculation**:
   Let's calculate these trigonometric values.

\[
x = \cos(251^\circ), \quad y = \sin(251^\circ)
\]

I'll compute the values now.The coordinates of the point \( P \) on the unit circle for the angle \( \alpha = 251^\circ \) are approximately:

\[
P \approx (-0.33, -0.95)
\]

Would you like more details or have any further questions?

Here are five related questions for deeper understanding:
1. What are the coordinates of a point on the unit circle for an angle of \( 90^\circ \)?
2. How do trigonometric values change for angles greater than \( 360^\circ \)?
3. Can you derive the same result using radians instead of degrees?
4. What happens to the sine and cosine values when the angle is negative?
5. How do sine and cosine values behave in different quadrants of the unit circle?

**Tip**: In the unit circle, angles between \( 180^\circ \) and \( 360^\circ \) correspond to negative \( x \)- or \( y \)-coordinates.

Auf dem Einheitskreis um den Punkt O = (0;0) liegt ein Punkt P so, dass die positive x-Achse und die in O beginnende Halbgerade durch P den Winkel α = 251° einschließen. Welche Koordinaten hat der Punkt P? Runden Sie das Ergebnis auf zwei Nachkommastellen.

This problem involves finding the coordinates of point P on the unit circle given an angle of α = 251°. Using trigonometric functions, the coordinates are calculated and rounded to two decimal places. Suitable for Grades 9-12.

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