Polar Coordinates
Coordinate systems are tools that let us use algebraic methods to understand geometry. While the rectangular (also called Cartesian) coordinates that we have been using are the most common, some problems are easier to analyze in alternate coordinate systems.
A coordinate system is a scheme that allows us to identify any point in the plane or in threedimensional space by a set of numbers. In rectangular coordinates these numbers are interpreted, roughly speaking, as the lengths of the sides of a rectangle. In polar coordinates a point in the plane is identified by a pair of numbers $(r,\theta)$. The number $\theta$ measures the angle between the positive $x$axis and a ray that goes through the point, as shown in figure 10.1.1; the number $r$ measures the distance from the origin to the point. Figure 10.1.1 shows the point with rectangular coordinates $\ds (1,\sqrt3)$ and polar coordinates $(2,\pi/3)$, 2 units from the origin and $\pi/3$ radians from the positive $x$axis.
Example 10.1.1 
SolutionJust as we describe curves in the plane using equations involving $x$ and $y$, so can we describe curves using equations involving $r$ and $\theta$. Most common are equations of the form $r=f(\theta)$. 
Example 10.1.1 

Example 10.1.2 

Each point in the plane is associated with exactly one pair of numbers in the rectangular coordinate system; each point is associated with an infinite number of pairs in polar coordinates. In the cardioid example, we considered only the range $0\le \theta\le2\pi$, and already there was a duplicate: $(2,0)$ and $(2,2\pi)$ are the same point. Indeed, every value of $\theta$ outside the interval $[0,2\pi)$ duplicates a point on the curve $r=1+\cos\theta$ when $0\le\theta < 2\pi$. We can even make sense of polar coordinates like $(2,\pi/4)$: go to the direction $\pi/4$ and then move a distance 2 in the opposite direction; see figure 10.1.3. As usual, a negative angle $\theta$ means an angle measured clockwise from the positive $x$axis. The point in figure 10.1.3 also has coordinates $(2,5\pi/4)$ and $(2,3\pi/4)$.
The relationship between rectangular and polar coordinates is quite easy to understand. The point with polar coordinates $(r,\theta)$ has rectangular coordinates $x=r\cos\theta$ and $y=r\sin\theta$; this follows immediately from the definition of the sine and cosine functions. Using figure 10.1.3 as an example, the point shown has rectangular coordinates $\ds x=(2)\cos(\pi/4)=\sqrt2\approx 1.4142$ and $\ds y=(2)\sin(\pi/4)=\sqrt2$. This makes it very easy to convert equations from rectangular to polar coordinates.
Example 10.1.3 

Example 10.1.4 

Example 10.1.5 

Converting polar equations to rectangular equations can be somewhat trickier, and graphing polar equations directly is also not always easy.
Example 10.1.6 

Exercises 10.1
Ex 10.1.1Plot these polar coordinate points on one graph: $(2,\pi/3)$, $(3,\pi/2)$, $(2,\pi/4)$, $(1/2,\pi)$, $(1,4\pi/3)$, $(0,3\pi/2)$.
Find an equation in polar coordinates that has the same graph as the given equation in rectangular coordinates.
Ex 10.1.2$\ds y=3x$ (answer)
Ex 10.1.3$\ds y=4$ (answer)
Ex 10.1.4$\ds xy^2=1$ (answer)
Ex 10.1.5$\ds x^2+y^2=5$ (answer)
Ex 10.1.6$\ds y=x^3$ (answer)
Ex 10.1.7$\ds y=\sin x$ (answer)
Ex 10.1.8$\ds y=5x+2$ (answer)
Ex 10.1.9$\ds x=2$ (answer)
Ex 10.1.10$\ds y=x^2+1$ (answer)
Ex 10.1.11$\ds y=3x^22x$ (answer)
Ex 10.1.12$\ds y=x^2+y^2$ (answer)
Sketch the curve.
Ex 10.1.13$\ds r=\cos\theta$
Ex 10.1.14$\ds r=\sin(\theta+\pi/4)$
Ex 10.1.15$\ds r=\sec\theta$
Ex 10.1.16$\ds r=\theta/2$, $\theta\ge0$
Ex 10.1.17$\ds r=1+\theta^1/\pi^2$
Ex 10.1.18$\ds r=\cot\theta\csc\theta$
Ex 10.1.19$\ds r={1\over\sin\theta+\cos\theta}$
Ex 10.1.20$\ds r^2=2\sec\theta\csc\theta$
\bigbreak Find an equation in rectangular coordinates that has the same graph as the given equation in polar coordinates.
Ex 10.1.21$\ds r=\sin(3\theta)$ (answer)
Ex 10.1.22$\ds r=\sin^2\theta$ (answer)
Ex 10.1.23$\ds r=\sec\theta\csc\theta$ (answer)
Ex 10.1.24$\ds r=\tan\theta$ (answer)