To solve the problem, we will follow these steps:

1. Parametric Equation of a Circle (Center at the Origin):

For a circle with radius $r$ and center at the origin $(0,0)$, the parametric equations are: $x = r \cos \theta$ $y = r \sin \theta$ where $\theta$ is the parameter (the angle), and it ranges from $0$ to $2\pi$.

2. Finding the Slopes at Specific Angles (0°, 45°, and 90°):

The slope of the tangent line to the circle at any point can be found by differentiating the parametric equations with respect to the angle $\theta$.

• Step 1: Parametric derivatives Differentiate $x = r \cos \theta$ and $y = r \sin \theta$ with respect to $\theta$: $\frac{dx}{d\theta} = -r \sin \theta$ $\frac{dy}{d\theta} = r \cos \theta$

• Step 2: Tangent slope formula The slope of the tangent to the circle is given by: $\text{slope} = \frac{dy/d\theta}{dx/d\theta} = \frac{r \cos \theta}{-r \sin \theta} = -\cot \theta$

3. Slopes at Specific Angles:

• At $\theta = 0^\circ$: $\text{slope} = -\cot(0^\circ) = 0$
• At $\theta = 45^\circ$: $\text{slope} = -\cot(45^\circ) = -1$
• At $\theta = 90^\circ$: $\text{slope} = -\cot(90^\circ) = 0$

Summary of Results:

• The parametric equations for the circle centered at the origin are: $x = r \cos \theta, \quad y = r \sin \theta$
• The slopes of the tangent at the angles $0^\circ$, $45^\circ$, and $90^\circ$ are: $\text{Slope at } 0^\circ = 0, \quad \text{Slope at } 45^\circ = -1, \quad \text{Slope at } 90^\circ = 0$

Would you like further clarification on any of the steps?

Here are some related questions to explore:

1. What happens to the parametric equations if the circle is shifted from the origin?
2. How would you compute the tangent slope at other angles, like 30° or 60°?
3. Can you develop parametric equations for ellipses or other conic sections?
4. How would the parametric equations change for a circle with a different radius?
5. How can the parametric equation be extended to 3D, such as for a sphere?

Tip: Parametric equations can be very useful for describing curves and geometric shapes beyond circles—try applying them to ellipses or spirals!