Rewrite the polar equation 
r
=
5
sin
(
θ
)
 as a Cartesian equation.

To rewrite the polar equation \( r = 5\sin(\theta) \) as a Cartesian equation, follow these steps:

1. **Recall the polar-to-Cartesian relationships**:
   - \( x = r\cos(\theta) \)
   - \( y = r\sin(\theta) \)
   - \( r^2 = x^2 + y^2 \)

2. **Express \( \sin(\theta) \) in terms of \( r \) and \( y \)**:
   - From \( \sin(\theta) = \frac{y}{r} \), substitute this into the given equation:
     \[
     r = 5\sin(\theta) \implies r = 5\frac{y}{r}.
     \]

3. **Eliminate \( r \) from the denominator**:
   - Multiply through by \( r \) (assuming \( r \neq 0 \)):
     \[
     r^2 = 5y.
     \]

4. **Replace \( r^2 \) with \( x^2 + y^2 \)**:
   - Substitute \( r^2 = x^2 + y^2 \) into the equation:
     \[
     x^2 + y^2 = 5y.
     \]

5. **Rearrange to standard Cartesian form**:
   - Rearrange the equation:
     \[
     x^2 + y^2 - 5y = 0.
     \]
   - Complete the square for the \( y \)-terms:
     \[
     x^2 + (y^2 - 5y) = 0 \implies x^2 + \left(y - \frac{5}{2}\right)^2 = \left(\frac{5}{2}\right)^2.
     \]

6. **Final Cartesian form**:
   - The equation becomes:
     \[
     x^2 + \left(y - \frac{5}{2}\right)^2 = \frac{25}{4}.
     \]

This represents a circle with center \( (0, \frac{5}{2}) \) and radius \( \frac{5}{2} \).

---

Would you like further clarification or details? Here are some questions to expand on this topic:

1. How do the polar-to-Cartesian relationships work in other equations?
2. Can you explain the geometric interpretation of \( r = 5\sin(\theta) \) in polar coordinates?
3. How do you complete the square, and why is it necessary here?
4. What are the differences between Cartesian and polar forms for representing circles?
5. Can you derive equations for other shapes, such as ellipses, from polar equations?

**Tip**: Completing the square is a crucial technique for rewriting quadratic equations into standard forms, especially for conics.

Rewrite the polar equation r = 5sin(θ) as a Cartesian equation.

Learn how to convert the polar equation r = 5sin(θ) into a Cartesian equation. This detailed solution explains key mathematical concepts such as polar and Cartesian coordinate relationships, completing the square, and conic sections. Ideal for high school students in Grades 10-12.

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Rewrite the polar equation r

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