Section 10.4 – Polar Coordinates and Polar Graphs.

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Presentation transcript:

Section 10.4 – Polar Coordinates and Polar Graphs

Introduction to Polar Curves Parametric equations allowed us a new way to define relations: with two equations. Parametric curves opened up a new world of curves: Polar coordinates will introduce a new coordinate system.

Introduction to Polar Curves You have only been graphing with standard Cartesian coordinates, which are named for the French philosopher-mathematician, Rene Descartes.

Polar Coordinates

Example 1

Example 2

Example 3 r 0

The Relationships Between Polar and Cartesian Coordinates Right triangles are always a convenient shape to draw. Using Pythagorean Theorem…

The Relationships Between Polar and Cartesian Coordinates You can use a reference angle to find a relationship but that would require an extra step. Instead, compare the coordinates to the unit circle coordinates.

The Relationships Between Polar and Cartesian Coordinates Therefore: (Remember tangent is also the slope of the radius.)

Conversion Between Polar and Cartesian Coordinates When converting between coordinate systems the following relationships are helpful to remember: NOTE: Because of conterminal angles and negative values of r, there are infinite ways to represent a Cartesian Coordinate in Polar Coordinates.

Example 1

Example 2 NOTE: In Cartesian coordinates, every point in the plane has exactly one ordered pair that describes it.

Example 3 A circle centered at (0,2) with a radius of 2 units.

Conversion Between Polar Equations and Parametric Equations Since we can easily convert a polar equation into parametric equations, the calculus for a polar equation can be performed with the parametrically defined functions. The slope of tangent lines is dy/dx not dr/dΘ.

Example Parametric Equations: Find dy/dx not dr/dΘ: Find the point: Find the equation:

Example (Continued) Parametric Equations: Find dy/dt and dx/dt: Use the Arc Length Formula:

Example (Continued) Parametric Equations: Find dy/dx: Find d 2 y/dx 2 : Since the second derivative is positive, the graph is concave up.

Alternate Formula for the Slope of a Tangent Line of a Polar Curve If you do not want to easily convert a polar equation into parametric equations, you can always memorize another formula...

Alternate Arc Length Formula for Polar Curves If you do not want to easily convert a polar equation into parametric equations, you can always memorize another formula...