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Notes

\({\text{Equations of Lines}}\)
\(\underline{\text{Polar Form}}\)	\(\underline{\text{Cartesian Form}}\)
\(\theta=\beta\)	\(y = (\tan \beta) x \)
\(r \cos \theta=a\)	\(x=a\)
\(r \sin \theta =b\)	\(y=b\)

\({\text{Equations of Circles}}\)
\(\underline{\text{Polar Form}}\)	\(\underline{\text{Cartesian Form}}\)
\(r=c\)	\(x^2+y^2=c^2\)
\(r= 2a\cos \theta\)	\((x-a)^2+y^2=a^2\)
\(r= 2b\sin \theta\)	\(x^2+(y-b)^2=b^2\)
\(r= 2a\cos \theta + 2b\sin \theta\)	\((x-a)^2+(y-b)^2=a^2+b^2\)

\({\text{Equations of Parabolas (Focus on the origin)}}\)
\(\underline{\text{Polar Form}}\)	\(\underline{\text{Cartesian Form}}\)
\(r=\frac{p}{1-\sin \theta}\)	\(y=\frac{1}{2p}x^2-\frac{p}{2}\)
\(r=\frac{p}{1+\sin \theta}\)	\(y=-\frac{1}{2p}x^2+\frac{p}{2}\)
\(r=\frac{p}{1-\cos \theta}\)	\(x=\frac{1}{2p}y^2-\frac{p}{2}\)
\(r=\frac{p}{1+\cos \theta}\)	\(x=-\frac{1}{2p}y^2+\frac{p}{2}\)

Questions & Videos

\(\textbf{1)}\) Convert the point from Polar to Rectangular Coordinates
\((8,60^{\circ})\)

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The answer is \((4,4\sqrt{3})\)

\(\textbf{2)}\) Convert the point from Polar to Rectangular Coordinates
\((2,180^{\circ})\)

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The answer is \((-2,0)\)

\(\textbf{3)}\) Convert the point from Rectangular to Polar Coordinates
\((-4,5)\)

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The answer is \((\sqrt{41},128.7^{\circ})\)

\(\textbf{4)}\) Convert the point from Rectangular to Polar Coordinates
\((0,-3)\)

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The answer is \((3,270^{\circ})\)

\(\textbf{5)}\) Convert the Polar Equation to Rectangular and identify the graph
\(r=2 \cos⁡ \theta\)

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The answer is \((x-1)^2+y^2=1, \,\,\) Circle

\(\textbf{6)}\) Convert the Polar Equation to Rectangular and identify the graph
\(r=\displaystyle \frac{5}{1-\cos \theta}\)

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The answer is \(y^2-10x-25=0, \,\,\) Parabola

\(\textbf{7)}\) Convert the Polar Equation to Rectangular and identify the graph
\(r=\displaystyle\frac{3}{3+ \sin \theta}\)

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The answer is \(9x^2+8y^2+6y=9, \,\,\) Ellipse

\(\textbf{8)}\) Convert the Polar Equation to Rectangular and identify the graph
\(r=5\)

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The answer is \(x^2+y^2=25, \,\,\) Circle

\(\textbf{9)}\) Convert the Polar Equation to Rectangular and identify the graph
\(r=6 \cos \theta\)

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The answer is \((x-3)^2+y^2=9, \,\,\) Circle

\(\textbf{10)}\) Convert the Polar Equation to Rectangular and identify the graph
\(r=-12 \sin \theta\)

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The answer is \(x^2+(y+6)^2=36, \,\,\) Circle

\(\textbf{11)}\) Convert the Polar Equation to Rectangular and identify the graph
\(r=6 \cos \theta – 8 \sin \theta\)

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The answer is \((x-3)^2+(y+4)^2=25, \,\,\) Circle

\(\textbf{12)}\) Convert the Polar Equation to Rectangular and identify the graph
\(\theta=\frac{\pi}{4}\)

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The answer is \(y=x, \,\,\) Line

\(\textbf{13)}\) Convert the Polar Equation to Rectangular and identify the graph
\(\theta=-\frac{\pi}{4}\)

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The answer is \(y=-x, \,\,\) Line

\(\textbf{14)}\) Convert the Polar Equation to Rectangular and identify the graph
\(r \cos \theta=2\)

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The answer is \(x=2, \,\,\) Vertical Line

\(\textbf{15)}\) Convert the Polar Equation to Rectangular and identify the graph
\(r \sin \theta=-3\)

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The answer is \(y=-3, \,\,\) Horizontal Line

\(\textbf{16)}\) Convert the Polar Equation to Rectangular and identify the graph
\(r=\frac{4}{1+ \sin \theta}\)

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The answer is \(y=-\frac{1}{8}x^2+2, \,\,\) Parabola

In Summary

Converting to polar coordinates is the process of expressing a point in the polar coordinate system \(\left(r , \theta \right) \) starting with the cartesian coordinate system \(\left(x , y \right) \). This involves using formulas that describe the relationship between the 2 coordinate types, \(x = r \cos \theta\) and \(y = r \sin \theta \). Being able to convert to polar coordinates is an important skill in geometry and trigonometry, and it has many applications in mathematics, physics, and engineering.

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