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) θ is 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. As usual, it’s easier to DO it than to explain it!
Polar Coordinates
As usual, it’s easier to DO it than to explain it!

20. Plot (2,135º) Begin with the angle θ = 135º:
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. Check Point 1 page 685 Plot the points: (3,315º) (–2,π) (–1,–π/2)
Polar Coordinates
IN-CLASS PROBLEMS
Check Point 1 page 685
Plot the points:
(3,315º)
(–2,π)
(–1,–π/2)

30. 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

31. 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?

32. Polar Coordinates
Rectangular to polar coordinates r = 𝒙 𝟐 + 𝒚 𝟐 θ = arctan (y ÷ x)

33. Polar Coordinates
IN-CLASS PROBLEMS
r = 𝒙 𝟐 + 𝒚 𝟐 θ = arctan (y ÷ x) Find the polar coordinates of: (x,y) = (1, – 𝟑 ) (x,y) = (0, – 4)

34. Questions?

35. Polar Equations
Polar equations have variables r and θ

36. Polar Equations
Converting rectangular equations to polar equations: replace x with r cosθ and y with r sinθ

37. Polar Equations
Converting polar equations to rectangular equations: try r 2 = x2 + y2 r cosθ = x r sinθ = y tanθ = y/x

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

39. 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

40. Polar Equations

41. Polar Equations

42. Polar Equations
Spiral of Archimedes r = aθ

43. Polar Equations

44. Polar Equations

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

46. Questions?

47. Complex Plane
Remember the imaginary unit i
i = −1

48. Complex Plane
Remember we didn’t allow any exponents when using i
i = −1
i 2 = -1

49. Complex Plane i = −1 i 2 = -1 i 3 = - i i 4 = 1 i 5 = i
You can keep on going:
i = −1
i 2 = -1
i 3 = - i
i 4 = 1
i 5 = i

50. But, that pattern formed a complete cycle, and you can keep cycling forever!
i 7 = - i
i 8 = 1
i 9 = i …

51. Complex Plane
Remember complex #s: z = a + bi

52. Complex Plane
real #s are points on the real # line

53. Complex Plane
complex numbers can be plotted as points on the “complex plane”: real axis (horizontal) and an imaginary axis (vertical)

54. Complex Plane
Check Point 1 page 707 Plot: a) z = 2 + 3i b) z = i c) z = -4 d) z = -i

55. Complex Plane
Fractals/Mandelbrot sets Graphing complex numbers

56. Complex Plane
Video: Fractals

57. Questions?

58. Using Complex Numbers
More complex numbers!

59. Using Complex Numbers
Remember the absolute value |x| is the distance between x and 0:

60. Using Complex Numbers
But absolute value is not what we usually use to calculate a distance because it only works for horizontal or vertical distances along an axis

61. Using Complex Numbers √
d = (y2 – y1)2 + (x2 – x1)2 is the formula used to calculate distances “d” in virtually all technical equations

62. Using Complex Numbers
This is based on the Pythagorean Theorem

63. Using Complex Numbers
So, an absolute value and a square root of a sum of squares are really both a measure of distance (and hence, the same thing)

64. Using Complex Numbers √
We can calculate the absolute value of a complex number z = a + bi as: |z| = |a + b i| = a2 + b2 √

65. Using Complex Numbers
Check Point 2 page 708 Find the absolute value of: a) z = i b) z = 2 - 3i

66. Using Complex Numbers
Polar form of a complex number: z = a + bi becomes: z = r (cos θ + i sin θ ) where a = r cos θ b = r sin θ r = a 2 + b 2 tan θ = b/a

67. Using Complex Numbers
r is called the modulus θ is called the argument

68. Using Complex Numbers
Always plot these first or you may end up in the wrong quadrant! (This is because of the tan)

69. Using Complex Numbers
Example 3 page 708 Plot z = -2 – 2i in the complex plane then write z in polar form

70. Using Complex Numbers
z = -2 – 2i z = a + bi a = ? b = ?

71. Using Complex Numbers
z = -2 – 2i z = a + bi a = -2 b = -2 Plot?

72. Using Complex Numbers
z = -2 – 2i
So it’s in quadrant 3
r = ?
Θ = ?

73. Using Complex Numbers z = -2 – 2i in quadrant 3 a = -2 b = -2
= 8 = 2 2
Θ = ?

74. Using Complex Numbers z = -2 – 2i in quadrant 3 a = -2 b = -2
tanΘ = b/a = -2/-2
So tan-1(1) = Θ

75. Using Complex Numbers z = -2 – 2i in quadrant 3 a = -2 b = -2
If tan-1(1) = Θ (use your table!)
then Θ = π/4 or 5π/4
Which?

76. Using Complex Numbers z = -2 – 2i in quadrant 3 a = -2 b = -2

77. Using Complex Numbers z = -2 – 2i Θ = 5π/4 SO what is:
z = r (cos θ + i sin θ)

78. Using Complex Numbers z = -2 – 2i Θ = 5π/4 SO:
z = (cos 5π/4 + i sin 5π/4)

80. Using Complex Numbers
Rectangular form of a complex number (You don’t have to plot these first!)

81. Using Complex Numbers
Example 4 on page 709 Write z = 2(cos(60o) + i sin(60o)) in rectangular form

82. z = 2(cos(60o) + i sin(60o)) would mean: r = 2 Θ = 60 o
Using Complex Numbers
z = 2(cos(60o) + i sin(60o)) would mean: r = 2 Θ = 60 o

83. Using Complex Numbers
Write z = 2(cos(60o) + i sin(60o)) r = 2 Θ = 60 o z = 2(1/2 + i 𝟑 /2) z = 1 + 𝟑 i

84. Questions?

85. DeMoivre’s Theorem
Powers of complex numbers in polar form

86. DeMoivre’s Theorem
the power of a complex number z = r (cos θ + i sin θ ) For n>0 zn = [r (cos θ + i sin θ )] n = rn(cos nθ + i sin nθ )

87. DeMoivre’s Theorem
Example 7 page 712 Find [2(cos 20° + i sin 20°)]6

88. DeMoivre’s Theorem
[2(cos 20° + i sin 20°)]6 zn = [r (cos θ + i sin θ )] n So, what is n? What is r? What is θ?

89. DeMoivre’s Theorem
[2(cos 20° + i sin 20°)]6 zn = [r (cos θ + i sin θ )] n n = 6 r = 2 θ = 20° yay! No radians!

90. DeMoivre’s Theorem
n = 6 r = 2 θ = 20° zn = rn(cos nθ + i sin nθ ) = 26(cos 6(20°) + i sin 6(20°))

91. DeMoivre’s Theorem
= 64 (cos 120° + i sin 120°) = 64(-1/2 + i 3 /2) = i 3

92. DeMoivre’s Theorem
Check Point 7 page 712 Find [2(cos 30° + i sin 30°)]5

93. DeMoivre’s Theorem
Example 8 page 712 Find (1 + i)8 z = a + bi a = ? b = ?

94. DeMoivre’s Theorem
Example 8 page 712 Find (1 + i)8 z = a + bi a = 1 b = 1

95. DeMoivre’s Theorem
(1 + i)8 a = 1 b = 1 Which quadrant?

96. DeMoivre’s Theorem
(1 + i)8 Quadrant 1 a = 1 b = 1 r = a 2 + b 2 = ?

97. DeMoivre’s Theorem
(1 + i)8 Quadrant 1 a = 1 b = 1 r = a 2 + b 2 = = 2 tan Θ = b/a = ?

98. DeMoivre’s Theorem
(1 + i)8 Quadrant 1 a = 1 b = 1 r = a 2 + b 2 = = 2 tan Θ = b/a = 1/1 = 1 and Θ = π/4 or 45° because Θ is in Quadrant 1

99. DeMoivre’s Theorem
(1 + i )8 r = a 2 + b 2 = = 2 Θ = π/4 or 45° 1 + i = r(cos Θ + i sin Θ) = 2 (cos π/4 + i sin π/4)

100. DeMoivre’s Theorem
Use DeMoivre to raise it to the power 8: (1 + i)8 = [ 2 (cos π/4 + i sin π/4)]8 = ( 2 )8(cos 8*π/4 + i sin 8*π/4) = 16(cos 2π + i sin 2π) = 16(1 + 0 i ) = i or 16

101. DeMoivre’s Theorem
Check Point 8 page 713 Find (1 + i )4

102. Questions?

103. Liberation!
Be sure to turn in your assignments from last week to me before you leave Don’t forget your homework due next week! Have a great rest of the week!