How do you convert $r = \frac{12}{4 + 8 sin x}$ $r = 12/(4 + 8 sinx)$ into a rectangular equation?

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Answer 1 · 2015-11-22T14:30:10

Use these 3 identities to solve polar equations:

1) $r = \sqrt{x^{2} + y^{2}}$ $r=sqrt(x^2+y^2)$
2) $x = r cos θ$ $x=rcostheta$
3) $y = r sin θ$ $y=rsintheta$

First simplify the polar equation:

$r = \frac{3}{1 + 2 sin θ}$ $r=3/(1+2sintheta)$

$r + 2 r sin θ = 3$ $r+2rsintheta=3$

Using the identities above, use substitution to eliminate the r and sine function:

$\sqrt{x^{2} + y^{2}} + (2 r) (\frac{y}{r}) = 3$ $sqrt(x^2+y^2)+(2r)(y/r)=3$

Now, simplify

$\sqrt{x^{2} + y^{2}} + 2 y = 3$ $sqrt(x^2+y^2)+2y=3$

$\sqrt{x^{2} + y^{2}} = 3 - 2 y$ $sqrt(x^2+y^2)=3-2y$

$x^{2} + y^{2} = {(3 - 2 y)}^{2} = 9 - 12 y + 4 y^{2}$ $x^2+y^2=(3-2y)^2=9-12y+4y^2$

Rectangular Equation : $3 y^{2} - x^{2} - 12 y + 9 = 0$ $3y^2-x^2-12y+9=0$

This is a hyperbola with a vertical transverse axis .

Hope that helped

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How do you convert r = 12/(4 + 8 sinx)r=124+8sinxr = 12/(4 + 8 sinx) into a rectangular equation?