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Section 10.4 – Polar Coordinates and Polar Graphs

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.

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Section 10.4 – Polar Coordinates and Polar Graphs

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  1. Section 10.4 – Polar Coordinates and Polar Graphs

  2. 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.

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

  4. Polar Coordinates In polar coordinates we identify the origin as the pole and the positive -axis as the polar axis. We can then identify each point in the plane by polar coordinates , where gives the distance from to and gives the angle from the initial ray to the ray . By convention, angles measured in the counterclockwise direction are positive. Since it easier to plot a point by starting with the angle, polar equations are like inverses. independent variable. dependent variable. NOTE: The origin has no well-defined coordinate. For our purposes the coordinates will be for any .

  5. Example 1 Example: Plot the polar coordinates . First find the angle on the polar grid. To plot a point using polar coordinates , we often use a polar grid: Now plot the point units in the direction of the angle.

  6. Example 2 Example: Plot the polar coordinates . First find the angle on the polar grid. To plot a point using polar coordinates , we often use a polar grid: Now plot the point units in the direction of the angle. If is negative, the point is plotted units in the opposite direction. 2

  7. Example 3 Graph the polar curve . Indicate the direction in which it is traced. Notice Polar equations are like inverses. independent variable. dependent variable.

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

  9. The Relationships Between Polar and Cartesian Coordinates Find the relationships between . What about the angle ? 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 red and blue triangles are similar with a scale factor of . Thus…

  10. The Relationships Between Polar and Cartesian Coordinates Find the relationships between . What about a relationship with ? To find the angle measure , it is possible to use the tangent function to find the reference angle. Instead investigate the tangent function and : Therefore: (Remember tangent is also the slope of the radius.)

  11. 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.

  12. Example 1 Complete the following: • Convert into polar coordinates. • Express your answer in (a) as many ways as you can.

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

  14. Example 3 Use the polar-rectangular conversion formulas to show that the polar graph of is a circle. A circle centered at (0,2) with a radius of 2 units.

  15. Conversion Between Polar Equations and Parametric Equations The polar graph of is the curve defined parametrically by: Example: Write a set of parametric equations for the polar curve The slope of tangent lines is dy/dx not dr/dΘ. 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.

  16. Example Use polar equation to answer the following questions: (a) Find the Cartesian equation of the tangent line at . Find the slope of the tangent line (Remember ): Parametric Equations: Find the point: Find dy/dx not dr/dΘ: Find the equation:

  17. Example (Continued) Use polar equation to answer the following questions: (b) Find the length of the arc from to . Parametric Equations: Use the Arc Length Formula: Find dy/dt and dx/dt:

  18. Example (Continued) Use polar equation to answer the following questions: (c) Is the curve concave up or down at . Parametric Equations: Find dy/dx: Find d2y/dx2: Since the second derivative is positive, the graph is concave up. Find value of the second derivative (Remember ):

  19. Alternate Formula for the Slope of a Tangent Line of a Polar Curve If is a differentiable function of , then the slope of the tangent line to the graph of at the point is: If you do not want to easily convert a polar equation into parametric equations, you can always memorize another formula...

  20. Alternate Arc Length Formula for Polar Curves The arc length for a polar curve between and is given by If you do not want to easily convert a polar equation into parametric equations, you can always memorize another formula...

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