convert the following: r =2cos +2sin      to rectanglar

To convert the polar equation \( r = 2\cos\theta + 2\sin\theta \) to rectangular form, we use the following relationships between polar and rectangular coordinates:

1. \( x = r\cos\theta \)
2. \( y = r\sin\theta \)
3. \( r = \sqrt{x^2 + y^2} \)
4. \( \cos\theta = \frac{x}{r} \), \( \sin\theta = \frac{y}{r} \)

### Steps to Convert:
The given equation is:
\[
r = 2\cos\theta + 2\sin\theta
\]

1. Multiply through by \( r \) to eliminate the fraction involving \( \cos\theta \) and \( \sin\theta \):
\[
r^2 = 2r\cos\theta + 2r\sin\theta
\]

2. Substitute \( r^2 = x^2 + y^2 \), \( r\cos\theta = x \), and \( r\sin\theta = y \):
\[
x^2 + y^2 = 2x + 2y
\]

3. Rearrange the equation:
\[
x^2 + y^2 - 2x - 2y = 0
\]

4. Complete the square for \( x \) and \( y \):
   - For \( x^2 - 2x \), add and subtract \( \left(\frac{2}{2}\right)^2 = 1 \).
   - For \( y^2 - 2y \), add and subtract \( \left(\frac{2}{2}\right)^2 = 1 \).

\[
(x^2 - 2x + 1) + (y^2 - 2y + 1) = 0 + 1 + 1
\]

5. Simplify to form perfect squares:
\[
(x - 1)^2 + (y - 1)^2 = 2
\]

### Final Answer:
The rectangular form of the equation is:
\[
(x - 1)^2 + (y - 1)^2 = 2
\]

This represents a circle with center \( (1, 1) \) and radius \( \sqrt{2} \).

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### Related Questions:
1. What is the general form of a circle's equation in rectangular coordinates?
2. How can you convert a polar equation involving \( r^2 \) directly to rectangular form?
3. Why do we use the substitution \( r^2 = x^2 + y^2 \) during this process?
4. How would the process change if the coefficients of \( \cos\theta \) and \( \sin\theta \) were different?
5. How can you determine the center and radius of a circle from its equation?

**Tip:** Completing the square is a fundamental technique in algebra, especially for converting conic sections into standard forms. Practice often to master it!

Convert the following: r = 2cosθ + 2sinθ to rectangular

Learn how to convert the polar equation r = 2cosθ + 2sinθ to its rectangular form with a step-by-step solution. This problem covers key concepts such as polar and rectangular coordinates, completing the square, and the Pythagorean theorem. Suitable for students in Grades 10-12.

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