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Welcome to Week 6 College Trigonometry

Welcome to Week 6 College Trigonometry. Polar Coordinates. We know about square graph paper. Polar Coordinates. Now we’re going to learn about circular graph paper!. Polar Coordinates.

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Welcome to Week 6 College Trigonometry

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  1. Welcome to Week 6 College Trigonometry

  2. Polar Coordinates We know about square graph paper

  3. Polar Coordinates Now we’re going to learn about circular graph paper!

  4. Polar Coordinates In the 700s and 800s AD, Arabic astronomers developed methods for calculating the direction and distance to Meccah from any point on Earth

  5. Polar Coordinates They were using spherical trigonometry to do this

  6. Polar Coordinates Just like with square graph paper, polar graph paper has a starting point . .

  7. Polar Coordinates For square graph paper, this point is called the origin For polar graph paper, it is called the pole . .

  8. Polar Coordinates Just like square graph paper, the starting position is the right horizontal line

  9. Polar Coordinates This is called the “polar axis”

  10. Polar Coordinates The rotation for both square and polar graphs is counterclockwise II I III IV

  11. Polar Coordinates Just like for (x,y) coordinates, there are a pair of polar coordinates (r,θ) These are called the “polar coordinates”

  12. Polar Coordinates r is the distance from the pole (the radius) θ is the angle around the circle

  13. Polar Coordinates Angles in polar notation can be expressed in either degrees or radians

  14. Polar Coordinates Degrees are traditionally used in navigation, surveying, and many applied disciplines Radians are more common in mathematics and mathematical physics

  15. Polar Coordinates point P = (r,θ) r is the distance (radius) from the pole to P(+,- or 0) θis the angle from the polar axis to the terminal side of the angle (degrees or radians)

  16. Polar Coordinates Positive angles are measured counterclockwise from the polar axis Negative angles are measured clockwise from the polar axis

  17. Polar Coordinates P = (r, θ) is located |r| units from the pole

  18. Polar Coordinates r > 0 – the point lies on the terminal side of θ r < 0 - point lies along the ray opposite the terminal side of θ r = 0 the point lies on the pole no matter what the value of θ is!

  19. Polar Coordinates As usual, it’s easier to DO it than to explain it!

  20. Polar Coordinates IN-CLASS PROBLEMS Plot (2,135º) Begin with the angle θ = 135º:

  21. Polar Coordinates IN-CLASS PROBLEMS Because the angle is positive, the point will be along this line It will be r=2 radii from the pole

  22. Polar Coordinates IN-CLASS PROBLEMS Plot (-3,270º)

  23. Polar Coordinates IN-CLASS PROBLEMS Plot (-3,270º)

  24. Polar Coordinates IN-CLASS PROBLEMS Plot (-3,270º)

  25. Polar Coordinates IN-CLASS PROBLEMS Plot (-1,-45º)

  26. Polar Coordinates IN-CLASS PROBLEMS Plot (-1,-45º)

  27. Polar Coordinates IN-CLASS PROBLEMS Plot (-1,-45º)

  28. Polar Coordinates IN-CLASS PROBLEMS Check Point 1 page 685 Plot the points: (3,315º) (–2,π) (–1,–π/2)

  29. Polar Coordinates Polar to rectangular conversion If you have a polar point P = (r,θ) To convert to (x,y) coordinates: x = r cos θ y = r sin θ

  30. Polar Coordinates IN-CLASS PROBLEMS x = r cos θ y = r sin θ Find the rectangular coordinates: a) P = (3,π) b) P = (-10,π/6) What quadrant is each in?

  31. Polar Coordinates Rectangular to polar coordinates r= θ= arctan (y ÷ x)

  32. Polar Coordinates IN-CLASS PROBLEMS r = θ = arctan (y ÷ x) Find the polar coordinates of: (x,y) = (1, –) Find the polar coordinates of: (x,y) = (0, – 4)

  33. Questions?

  34. Polar Equations Polar equations have variables r and θ

  35. Polar Equations Converting rectangular equations to polar equations: replace xwith rcosθand ywith rsinθ

  36. Polar Equations Converting polar equations to rectangular equations: try r 2 = x2 + y2 rcosθ= x rsinθ= y tanθ= y/x

  37. Polar Equations This is not easy – you may have to square both sides, take the tangent of both sides, multiply both sides by r

  38. Polar Equations People actually use polar equations for real work… But mostly you graph them because they change a ho-hum rectangular graph to a really interesting polar graph

  39. Polar Equations

  40. Polar Equations

  41. Polar Equations Spiral of Archimedes r = aθ

  42. Polar Equations

  43. Polar Equations

  44. Polar Equations Video: Polar Graphs graph of r = cos(2θ)

  45. Questions?

  46. Complex Plane Remember the imaginary unit i i=

  47. Complex Plane Remember we didn’t allow any exponents when using i i = i2= -1

  48. Complex Plane You can keep on going: i = i2= -1 i3= -i i4= 1 i5= i

  49. But, that pattern formed a complete cycle, and you can keep cycling forever! i5 = i6= -1 i7= -i i8= 1 i9= i …

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