11.1 Polar Coordinates and Graphs

Where is it? Coordinate systems are used to locate the position of a point. (3,1) (4,/6) In rectangular coordinates: We break up the plane into a grid of horizontal and vertical line lines. We locate a point by identifying it as the intersection of a vertical and a horizontal line. In polar coordinates: We break up the plane with circles centered at the origin and with rays emanating from the origin. We locate a point as the intersection of a circle and a ray.

(r, ) Polar axis The center of the graph is called the pole. Angles are measured from the positive x axis. Points are represented by a radius and an angle radius angle (r, ) To plot the point Polar axis First find the angle Then move out along the terminal side 5

A negative angle would be measured clockwise like usual. To plot a point with a negative radius, find the terminal side of the angle but then measure from the pole in the negative direction of the terminal side.

Let's plot the following points: Notice unlike in the rectangular coordinate system, there are many ways to list the same point.

Let's take a point in the rectangular coordinate system and convert it to the polar coordinate system. (3, 4) Based on the trig you know can you see how to find r and ? r 4  3 r = 5 We'll find  in radians (5, 0.93) polar coordinates are:

Let's generalize this to find formulas for converting from rectangular to polar coordinates. (x, y) r y  x

Now let's go the other way, from polar to rectangular coordinates. Based on the trig you know can you see how to find x and y? 4 y x rectangular coordinates are:

Let's generalize the conversion from polar to rectangular coordinates. y x

(8, 210°) (8, 210°) (6, -120°) (-5, 300°) (-5, 300°) (-5, 300°) Polar coordinates can also be given with the angle in degrees. (8, 210°) (8, 210°) (6, -120°) (-5, 300°) (-5, 300°) (-5, 300°) (-3, 540°) (-3, 540°) (-3, 540°)

Try plotting the following points in polar coordinates and find three additional polar representations of the point: Make sure to go over “sliding” through the pole to opposite ray when negative r is used.

Graphing a Polar Equation circle Note that some points are “retraced” as you mark them.

R = 2 + 2 sin t

R = 2+3cos(ϴ)

R = 4 cos 2t

Hwk. Pg. 400 1, 3-9, 11