MHB How to Convert Polar to Rectangular Coordinates in Calculus?

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To convert polar coordinates to rectangular coordinates for the equation r = 3 - cos(θ), the relationship r² = x² + y² is essential. The equation can be manipulated to x² + y² = 3r - x, leading to x² + y² = 3√(x² + y²) - x. This results in the equation (x² + x + y²)² = 9(x² + y²), which can be complex to simplify. The discussion highlights the challenges in handling the conversion, emphasizing that intuition can sometimes guide correct answers. Overall, the conversion process can be intricate and requires careful algebraic manipulation.
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$r=3-\cos\left({\theta}\right)$
${r}^{2}=3r-r\cos\left({\theta}\right)$
${x}^{2}+{y}^{2}=3r+x$
How you deal with 3r ?
 
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karush said:
$r=3-\cos\left({\theta}\right)$
${r}^{2}=3r-r\cos\left({\theta}\right)$
${x}^{2}+{y}^{2}=3r+x$
How you deal with 3r ?
You have a slight mistake: [math]x^2 + y^2 = 3r - x[/math]

As always [math]r = \sqrt{x^2 + y^2}[/math].

Continuing:
[math]x^2 + y^2 = 3 \sqrt{x^2 + y^2} - x[/math]

[math]x^2 + x + y^2 = 3 \sqrt{x^2 + y^2}[/math]

[math]\left ( x^2 + x + y^2 \right ) ^2 = 9(x^2 + y^2)[/math]

etc.

Yes, it's ugly.

-Dan
 
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Funny I had that answer and thot it was wrong, quess intuition is always right?
 
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