P OLAR E QUATIONS Section 10-4

Polar Coordinates Given: r: Directed distance from the Polar axis (pole) to point P Ɵ: Directed angle from the Polar axis to ray OP O Initial ray

Each point P in the plane can be assigned polar coordinates (r, Ɵ), as follows. r = directed distance from O to P Ɵ = directed angle, counterclockwise from polar axis to segment Polar Coordinates

Polar Graphs 1) Graph the following polar coordinates:

In general, the point (r, Ɵ) can be written as (r, Ɵ) = (r, Ɵ + 2nπ) or (r, Ɵ) = (–r, Ɵ + (2n + 1)π) where n is any integer. Moreover, the pole is represented by (0, Ɵ), where Ɵ is any angle. Polar Coordinates

To establish the relationship between polar and rectangular coordinates, let the polar axis coincide with the positive x-axis and the pole with the origin Because (x, y) lies on a circle of radius r, it follows that r 2 = x 2 + y 2. Coordinate Conversion

2) Convert to rectangular coordinates

3) Convert to rectangular coordinates 4) Convert to Polar coordinates

5) Convert the polar equation to rectangular form 6) Convert the polar equation to rectangular form

7) Convert the rectangular equation to polar form 8) Convert the rectangular equation to polar form

The graph of r = a is a circle of radius a centered at zero Ɵ = α is a Line through O making angle α with the initial ray Polar Graphs

Symmetric about the x-axis: if the point (r, Ɵ) lies on the graph, the point (-r, Ɵ) or (r, π-Ɵ) lies on the graph Symmetric about the y-axis: if the point (r, Ɵ) lies on the graph, the point (-r, -Ɵ) or (r, π-Ɵ) lies on the graph Symmetric about the origin: if the point (r, Ɵ) lies on the graph, the point (-r, Ɵ) or (r, π+Ɵ) lies on the graph Symmetry

9) Graph without a graphing calculator with the values of Ɵ from 0 to 2π. This curve is called a cardioid. To plot points use Polar Graphs

The following are simpler in polar form than in rectangular form. The polar equation of a circle having a radius of a and centered at the origin is simply Special Polar Graphs

Spiral of Archimedes

Using the parametric form of dy/dx we have Slope and Tangent Lines

Horizontal Vertical Horizontal and Vertical Tangent Lines Cusp at (0, 0)

If then Then the line Is tangent to the pole to the graph of Tangent Lines at the Pole

10) Find the equation of the line tangent to the polar curve

11) Find the vertical and horizontal tangents fo

H OME W ORK Worksheet 10-4