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Polar Coordinates

Polar Coordinates. Lesson 10.5. •. θ. r. Points on a Plane. (x, y). •. (r, θ ). Rectangular coordinate system Represent a point by two distances from the origin Horizontal dist, Vertical dist Also possible to represent different ways

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Polar Coordinates

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  1. Polar Coordinates Lesson 10.5

  2. • θ r Points on a Plane (x, y) • (r, θ) • Rectangular coordinate system • Represent a point by two distances from the origin • Horizontal dist, Vertical dist • Also possible to represent different ways • Consider using dist from origin, angle formed with positive x-axis

  3. Plot Given Polar Coordinates • Locate the following

  4. Find Polar Coordinates • A • A = • B = • C = • D = • B • D • C What are the coordinates for the given points?

  5. Converting Polar to Rectangular • r y θ x • Given polar coordinates (r, θ) • Change to rectangular • By trigonometry • x = r cos θy = r sin θ • Try = ( ___, ___ )

  6. Converting Rectangular to Polar • r y θ x • Given a point (x, y) • Convert to (r, θ) • By Pythagorean theorem r2 = x2 + y2 • By trigonometry • Try this one … for (2, 1) • r = ______ • θ = ______

  7. Polar Equations Note: for (r, θ) It is θ (the 2nd element that is the independent variable θ in degrees • States a relationship between all the points (r, θ) that satisfy the equation • Example r = 4 sin θ • Resulting values

  8. Graphing Polar Equations • Set Mode on TI calculator • Mode, then Graph => Polar • Note difference of Y= screen

  9. Graphing Polar Equations Also best to keepangles in radians Enter function in Y= screen

  10. Graphing Polar Equations • Set Zoom to Standard, • then Square

  11. Try These! • For r = A cos Bθ • Try to determine what affect A and B have • r = 3 sin 2θ • r = 4 cos 3θ • r = 2 + 5 sin 4θ

  12. Finding dy/dx • We know • r = f(θ) and y = r sin θ and x = r cos θ • Then • And

  13. Finding dy/dx • Since • Then

  14. Example • Given r = cos 3θ • Find the slope of the line tangent at (1/2, π/9) • dy/dx = ? • Evaluate •

  15. Define for Calculator It is possible to define this derivative as a function on your calculator

  16. Try This! • Find where the tangent line is horizontal for r = 2 cos θ • Find dy/dx • Set equal to 0, solve for θ

  17. Assignment Lesson 10.4 Page 736 Exercises 1 – 19 odd, 23 – 26 all Exercises 69 – 91 EOO

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