MATH 152 Chapter 13.4

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Polar Coordinates

Polar Coordinates

(back to 2 dimensions)

Cartesian to Polar

Let (x, y) be a 2D point. Define the polar coordinates of the point as follows: (r, θ)

$r=\pm {\sqrt {x^{2}+y^{2}}}$ (notice two representations)

$\tan {\theta }={\frac {y}{x}}$ (θ determined by quadrant)

Polar to Cartesian

$x=r\cos {\theta }$

$y=r\sin {\theta }$

Graphs of Polar Equations

r=sin(2θ)

$r=\sin {\left(2\theta \right)}$

Figure out what r is doing in each "quadrant" (quarter-period)

Period of $\sin {2\theta }$ is $\pi$

θ ∈	r ∈
[ 0, π/4 ]	[ 0, 1 ]
[ π/4, π/2 ]	[ 1, 0 ]
[ π/2, 3π/4 ]	[ 0, -1 ]
[ 3π/4, π ]	[ -1, 0 ]

Example

Sketch the region of the plane consisting of all points where 2 ≤ $r$ ≤ 4 and ${\tfrac {3\pi }{4}}\leq \theta \leq \pi$ and find the area of this region.

Area = ${\frac {1}{8}}\pi \left(4\right)^{2}-{\frac {1}{8}}\pi \left(2\right)^{2}$

Note: area of sector with radius r and central angle of θ is ${\frac {\theta }{2\pi }}\left(\pi {r}^{2}\right)={\frac {1}{2}}\pi {r}^{2}\theta$

Example

Sketch the polar curve $r=1+3\cos {\theta }$ and find a Cartesian equation for the curve.

Period = 2π

θ ∈	r ∈
[ 0, π/2 ]	[ 4, 1 ]
[ π/2, π ]	[ 1, –2 ]
[ π, 3π/2 ]	[ –2, 1 ]
[ 3π/2, 2π ]	[ 1, 4 ]

${\begin{aligned}r^{2}&=r+3r\cos {\theta }\\x^{2}+y^{2}&={\sqrt {x^{2}+y^{2}}}+3x\end{aligned}}$