Polar Coordinates

We have the following relationships between polar coordinates and rectangular coordinates:

$x=r\cos\theta$ $r^2=x^2+y^2$

$y=r\sin\theta$ $\tan\theta=\frac{y}{x}$

Ex 1 Find a set of polar coordinates for the point with rectangular coordinates $(5,5\sqrt{3})$.

Sol $r^2=x^2+y^2=5^2+(5\sqrt{3})^2=25+75=100$, so we can take $r=10$. Since $\tan\theta=y/x=(5\sqrt{3})/5=\sqrt{3}$, we can take $\theta=\pi/3$. Therefore $(10,\pi/3)$ is a set of polar coordinates for this point.

Ex 2 Find rectangular coordinates for the point which has polar coordinates $(6, 5\pi/6)$.

Sol $x=r\cos\theta=6\cos 5\pi/6=6(-\sqrt{3}/2)=-3\sqrt{3}$ and $y=r\sin\theta=6\sin 5\pi/6=6(1/2)=3$, so the point has rectangular coordinates $(-3\sqrt{3},3)$.

Pr A Find a polar equation for the circle with rectangular equation $x^2+y^2=25$.

Pr B Find the standard form of the rectangular equation of the circle with polar equation $r=4\cos\theta$.

Pr C Find two sets of polar coordinates for the point with rectangular coordinates $(-3,0)$.

Pr 1 Find a set of polar coordinates for the point with rectangular coordinates $(4\sqrt{3},-4)$.

Pr 2 Find a set of polar coordinates for the point with rectangular coordinates $(-5,-5)$.

Pr 3 Find rectangular coordinates for the point which has polar coordinates $(12, 4\pi/3)$.

Pr 4 Find a polar equation for the line with equation $x=4$ in rectangular coordinates.

Pr 5 Find a polar equation for the line with equation $y=6$ in rectangular coordinates.

Pr 6 Find a polar equation for the circle with equation $(x-3)^2+y^2=9$ in rectangular coordinates.

Pr 7 Find a rectangular equation for the cardioid with polar equation $r=5(1-\sin\theta)$.

Pr 8 Find the points of intersection of the curves with polar equations $r=6\cos\theta$ and $r=2+2\cos\theta$.

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