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

Polar Coordinates They were using spherical trigonometry to do this

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

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

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

Polar Coordinates This is called the “polar axis”

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

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

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

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

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

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)

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

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

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!

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!

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

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

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

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

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

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

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

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

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)

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

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?

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

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)

Questions?

Polar Equations Polar equations have variables r and θ

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

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

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

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

Polar Equations

Polar Equations

Polar Equations Spiral of Archimedes r = aθ

Polar Equations

Polar Equations

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

Questions?

Complex Plane Remember the imaginary unit i i = −1

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

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

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

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

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

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

Complex Plane Check Point 1 page 707 Plot: a) z = 2 + 3i b) z = -3 - 5i c) z = -4 d) z = -i

Complex Plane Fractals/Mandelbrot sets Graphing complex numbers

Complex Plane Video: Fractals

Questions?

Using Complex Numbers More complex numbers!

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

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

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

Using Complex Numbers This is based on the Pythagorean Theorem

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)

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

Using Complex Numbers Check Point 2 page 708 Find the absolute value of: a) z = 5 + 12i b) z = 2 - 3i

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

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

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

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

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

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

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

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

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

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?

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

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

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

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

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

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

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

Questions?

DeMoivre’s Theorem Powers of complex numbers in polar form

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 )

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

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?

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!

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

DeMoivre’s Theorem = 64 (cos 120° + i sin 120°) = 64(-1/2 + i 3 /2) = -32 + 32 i 3

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

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

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

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

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

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

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

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

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 ) = 16 + 0i or 16

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

Questions?

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!