https://file.aiquickdraw.com/mathbot/file/1727274613551ty3avwo5.jpg

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!

Develop a parametric equation of a circle in which the center lies on origin. Also determine the slopes at angle 0, 45, and 90 degrees.

Learn how to develop the parametric equation for a circle with its center at the origin. Explore how to calculate the slopes of the tangent at angles 0°, 45°, and 90° using the parametric derivatives and cotangent function. Suitable for advanced high school students (Grades 11-12).

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation