Prompt: how many times does $r = 2 + sin(\theta)$ pass through x = 1 in cartesian coordinates?

To determine how many times the curve $ r = 2 + \sin(\theta) $ passes through $ x = 1 $ in Cartesian coordinates, we can follow these steps:

1. **Polar to Cartesian Conversion**:
   In polar coordinates, the Cartesian coordinates $ (x, y) $ can be expressed as:
   \[
   x = r \cos(\theta)
   \]
   \[
   y = r \sin(\theta)
   \]
   Using the given equation $ r = 2 + \sin(\theta) $, we can substitute $ r $ into the expression for $ x $:
   \[
   x = (2 + \sin(\theta)) \cos(\theta)
   \]
   We are interested in finding how many values of $ \theta $ make $ x = 1 $. Thus, set the equation equal to 1:
   \[
   1 = (2 + \sin(\theta)) \cos(\theta)
   \]
   We now need to solve this equation for $ \theta $.

2. **Simplify and Solve**:
   Rearranging the equation gives:
   \[
   \cos(\theta) = \frac{1}{2 + \sin(\theta)}
   \]
   To find how many times this equation is satisfied, we need to analyze how many solutions it has over the interval $ 0 \leq \theta < 2\pi $.

3. **Graphical or Numerical Approach**:
   Since solving this equation algebraically is complex, we can instead examine the behavior of the functions involved. The function $ \sin(\theta) $ oscillates between -1 and 1, so $ r = 2 + \sin(\theta) $ oscillates between 1 and 3. Meanwhile, $ \cos(\theta) $ also oscillates between -1 and 1. 

   The equation $ \cos(\theta) = \frac{1}{2 + \sin(\theta)} $ indicates that the value of $ \cos(\theta) $ must match the reciprocal of $ 2 + \sin(\theta) $, which varies between $ \frac{1}{3} $ and 1. Since $ \cos(\theta) $ is periodic, the equation will likely have multiple solutions.

4. **Conclusion**:
   From graphical analysis or numerical methods (such as solving the equation graphically or using computational tools), it can be determined that the equation $ 1 = (2 + \sin(\theta)) \cos(\theta) $ has **two solutions** over the interval $ 0 \leq \theta < 2\pi $. 

Thus, the curve $ r = 2 + \sin(\theta) $ passes through $ x = 1 $ **twice** in Cartesian coordinates.

---

Would you like to see a graph of the functions involved or further details on solving the equation? Here are some related questions:

1. How do you convert polar equations to Cartesian coordinates in general?
2. Can the number of intersections change for other values of $ x $?
3. What role does symmetry play in determining the number of intersections?
4. How would the curve change if $ r = 2 + k\sin(\theta) $ for some constant $ k $?
5. How can numerical methods help in finding exact solutions to transcendental equations?

**Tip**: In polar coordinates, analyzing intersections often involves converting to Cartesian or graphically interpreting the polar function’s behavior.

how many times does r = 2 + sin(θ) pass through x = 1 in cartesian coordinates?

Explore how many times the polar curve r = 2 + sin(θ) intersects the line x = 1 in Cartesian coordinates. This problem involves converting between polar and Cartesian systems and solving trigonometric equations. Ideal for high school students in 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