9.3 Polar Coordinates 9.4 Areas and Lengths in Polar Coordinates

One way to give someone directions is to tell them to go three blocks East and five blocks South. Another way to give directions is to point and say “Go a half mile in that direction.” Polar graphing is like the second method of giving directions. Each point is determined by a distance and an angle. Initial ray A polar coordinate pair determines the location of a point. Polar Coordinates

To define the Polar Coordinates of a plane we need first to fix a point which will be called the Pole (or the origin) and a half-line starting from the pole. This half-line is called the Polar Axis. θ r P(r, θ) Polar Axis Polar Angles The Polar Angle θ of a point P, P ≠ pole, is the angle between the Polar Axis and the line connecting the point P to the pole. Positive values of the angle indicate angles measured in the counterclockwise direction from the Polar Axis. Polar Coordinates The Polar Coordinates (r,θ) of the point P, P ≠ pole, consist of the distance r of the point P from the Pole and of the Polar Angle θ of the point P. Every (0, θ) represents the pole. A positive angle.

More than one coordinate pair can refer to the same point. All of the polar coordinates of this point are:

The connection between Polar and Cartesian coordinates θ r (x,y)(x,y) y x From the right angle triangle in the picture one immediately gets the following correspondence between the Cartesian Coordinates (x,y) and the Polar Coordinates (r,θ) assuming the Pole of the Polar Coordinates is the Origin of the Cartesian Coordinates and the Polar Axis is the positive x-axis. x = r cos(θ) y = r sin(θ) r 2 = x 2 + y 2 tan(θ) = y/x Using these equations one can easily switch between the Cartesian and the Polar Coordinates.

Polar Curves Definition A Polar Curve consists of all the points (r,θ) satisfying a given equation F(r,θ) = 0. Often one can solve r form the equation and represent the polar curve in the form r = f(θ) Some curves are easier to describe with polar coordinates: (Circle centered at the origin) (Line through the origin)

Symmetry with respect to x-axis: If (r,  ) is on the graph, so is (r, -  ). Examples of Polar Curves

Symmetry with respect to y-axis: If (r,  ) is on the graph, so is (r,  -  )or (-r, -  ). Examples of Polar Curves

Areas in Polar Coordinates

Example: Find the area enclosed by:

Remember: For polar graphs: If we find derivatives and plug them into the formula, we (eventually) get: Arc Length in Polar Coordinates