In this example, the parameter $u$ represents a variable used to trace points along the circle. The angles (0°, 45°, and 90°) are mapped to different values of $u$ because the parameterization of the circle involves using $u$ to determine positions along the circumference.

1. Circle Parametric Equation: The parametric equations for a circle with radius $R$ and centered at the origin are given by: $x = R \cos(2 \pi u), \quad y = R \sin(2 \pi u), \quad 0 \leq u \leq 1$ This describes a complete rotation around the circle as $u$ goes from 0 to 1.

2. Mapping of Angles to Parameter $u$:

   • At 0°, which corresponds to the point $(R, 0)$, $u = 0$.
   • At 45°, the point lies on the first quadrant (i.e., at an angle halfway between 0° and 90°). This corresponds to $u = 0.125$, since: $u = \frac{45°}{360°} = 0.125$
   • At 90°, which corresponds to the point $(0, R)$, $u = 0.25$, since: $u = \frac{90°}{360°} = 0.25$

Therefore, the parameter $u$ is a fraction of the full revolution of the circle, where $u = 0$ is 0°, $u = 0.125$ is 45°, and $u = 0.25$ is 90°.

The tangent vectors at these points are computed by differentiating the parametric equation with respect to $u$, and then the slopes at these angles are calculated using the ratio $\frac{y'}{x'}$, which gives:

• 0° (u = 0): Slope = $\infty$ (vertical slope).
• 45° (u = 0.125): Slope = $-1$ (diagonal).
• 90° (u = 0.25): Slope = 0 (horizontal slope).

Let me know if you want further clarification or details on this, and here are some related questions:

1. How would the parametric equation change if the radius was different?
2. What happens to the tangent vector if $u = 0.5$ (180°)?
3. How does the slope change for angles greater than 90°?
4. Can we generalize the parametric equation for an ellipse?
5. How do the tangent vectors behave at other key points like $u = 0.75$ (270°)?

Tip: The parameter $u$ typically represents the fraction of the full angle (out of 360°), so dividing an angle by 360° gives the corresponding $u$-value for parametric equations involving circles.