Determine whether the equation is symmetric with respect to the x axis, y axis, origin, or a combination. r=4sec(theta)

To determine the symmetry of the polar equation \( r = 4 \sec(\theta) \), we can analyze it with respect to the x-axis, y-axis, and origin.

1. **Convert to Cartesian Coordinates**:
   We know:
   \[
   r \cos(\theta) = x \quad \text{and} \quad r \sin(\theta) = y
   \]
   Since \( \sec(\theta) = \frac{1}{\cos(\theta)} \), we can rewrite the equation as:
   \[
   r = \frac{4}{\cos(\theta)}
   \]
   Multiplying both sides by \( \cos(\theta) \) gives:
   \[
   r \cos(\theta) = 4
   \]
   Since \( r \cos(\theta) = x \), we have:
   \[
   x = 4
   \]
   This is a vertical line at \( x = 4 \).

2. **Analyze Symmetry**:
   - **Symmetry with respect to the x-axis**: Substitute \( (r, \theta) \) with \( (r, -\theta) \). The equation remains \( r = 4 \sec(\theta) \), which is unchanged. So, it is symmetric with respect to the x-axis.
   - **Symmetry with respect to the y-axis**: Substitute \( (r, \theta) \) with \( (-r, \pi - \theta) \). The equation does not hold in this case, so it is **not symmetric with respect to the y-axis**.
   - **Symmetry with respect to the origin**: Substitute \( (r, \theta) \) with \( (-r, \theta + \pi) \). The equation is not satisfied, so it is **not symmetric with respect to the origin**.

### Conclusion:
The equation \( r = 4 \sec(\theta) \) is symmetric only with respect to the x-axis.

Would you like further details on any steps or have any questions?

---

Here are some related questions for further study:
1. How do we determine symmetry for more complex polar equations?
2. What are the standard steps for converting between polar and Cartesian equations?
3. How does symmetry analysis differ between Cartesian and polar coordinates?
4. What are common symmetries seen in polar graphs, and how are they useful?
5. How can symmetry be used to simplify graphing polar functions?

**Tip**: When checking symmetry in polar coordinates, remember that substitutions for x-axis symmetry use \((r, -\theta)\), y-axis symmetry uses \((-r, \pi - \theta)\), and origin symmetry uses \((-r, \theta + \pi)\).

Analyze the symmetry of the polar equation r = 4sec(θ) to determine its symmetry with respect to the x-axis, y-axis, and origin. This solution includes conversions to Cartesian coordinates and checks for symmetry in polar coordinates. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation