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Cartesian vs. Polar

Cartesian vs. Polar. Relationship…. …using Pythagorean Theorem: r 2 = x 2 + y2. The Polar Coordinate System. Based on A point called the pole (often the origin) A ray called the polar axis , usually drawn in the direction of the positive x -axis.

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Cartesian vs. Polar

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  1. Cartesian vs. Polar

  2. Relationship… …using Pythagorean Theorem: r2 = x2 + y2

  3. The Polar Coordinate System • Based on • A point called the pole (often the origin) • A ray called the polar axis, usually drawn in the direction of the positive x-axis. The ordered pair P(r,  )gives the polar coordinates of point P.

  4. Plotting Points with Polar Coordinates Example Plot the point by hand in the polar coordinate system. Then determine the rectangular coordinates of each point.

  5. Negative r value… (b) Since r is –4, Q is 4 units in the negative direction from the pole on an extension of the ray. The rectangular coordinates

  6. Negative q value … (c) Since  is negative, the angle is measured in the clockwise direction. The rectangular coordinates

  7. Formulas Rectangular and Polar Coordinates If a point has rectangular coordinates (x, y) and polar coordinates (r,  ), then these coordinates are related as follows.

  8. Giving Alternative Forms for Coordinates of a Point Example • Give three other pairs of polar coordinates for the point P(3, 140º). • Determine two pairs of polar coordinates for the point with rectangular coordinates (–1, 1). Solution • See the figure: (3, –220º), (–3, 320º), and (–3, –40º).

  9. Classifying Polar Equations

  10. Graphing a Polar Equation (Cardioid) Graphing Calculator Solution Under the mode key choose DEGREE and POL to graph a polar equation, graph it for  in the interval [0º, 360º].

  11. Converting a Polar Equation to a Rectangular One Example For the polar equation • convert to a rectangular equation, • use a graphing calculator to graph the polar equation for 0    2, and • use a graphing calculator to graph the rectangular equation. Solution Multiply both sides by the denominator.

  12. Converting a Polar Equation to a Rectangular One Square both sides. Rectangular equation

  13. Converting a Polar Equation to a Rectangular One • The figure shows a graph with polar coordinates. • Solving x2 = –8(y – 2) for y, we obtain

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