How do I convert the polar equation $r = 10 sin θ$ $r=10 sin theta$ to its Cartesian equivalent?

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How do I convert the polar equation $r = 10 sin θ$ $r=10 sin theta$ to its Cartesian equivalent?

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Answer 1 · 2014-10-02T21:09:10

Cartesian coordinates, also known as Rectangular coordinates, are defined in terms of $x$ $x$ and $y$ $y$ . So, for this problem $θ$ $theta$ has to be eliminated/converted using basic foundations described by the unit circle and right triangle trigonometry.

$r = 10 sin (θ)$ $r=10sin(theta)$

Remember that ...

$x = r \cdot cos (θ)$ $x=r*cos(theta)$

$y = r \cdot sin (θ)$ $y=r*sin(theta)$

$r^{2} = x^{2} + y^{2}$ $r^2=x^2+y^2$

Multiply both sides of the equation by $r$ $r$

$r \cdot r = 10 r \cdot sin (θ)$ $r*r=10r*sin(theta)$

$r^{2} = 10 r \cdot sin (θ)$ $r^2=10r*sin(theta)$

$x^{2} + y^{2} = 10 r \cdot sin (θ)$ $x^2+y^2=10r*sin(theta)$

Use the fact that $y = r \cdot sin (θ)$ $y=r*sin(theta)$ to make a substitution.

$x^{2} + y^{2} = 10 y$ $x^2+y^2=10y$

$x^{2} + y^{2} - 10 y = 0$ $x^2+y^2-10y=0$

The above equation is the Cartesian/Rectangular coordinate equivalent to the given Polar equation.

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How do I convert the polar equation $r = 10 sin θ$ $r=10 sin theta$ to its Cartesian equivalent?

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