Download Unit06 PowerPoint for trigonometry class

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
III
I
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!
Polar Coordinates
As usual, it’s
easier to DO it
than to explain
it!
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)
Polar Coordinates
IN-CLASS PROBLEMS
Check Point 1 page 685
Plot the points:
a) (3,315º)
b) (–2,π)
c) (–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 =
𝒙𝟐 + 𝒚𝟐
θ = arctan (y ÷ x)
Polar Coordinates
IN-CLASS PROBLEMS
r =
𝒙𝟐 + 𝒚𝟐
θ = arctan (y ÷ x)
Find the polar coordinates of:
(x,y) = (1, – 𝟑)
Find the polar coordinates of:
(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 = x 2 + 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
You can keep on going:
i = −1
i 2 = -1
i3 = - i
i4 = 1
i5 = i
But, that pattern formed a
complete cycle, and you can
keep cycling forever!
i 5 = −1
i 6 = -1
i7 = - i
i8 = 1
i9 = 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| =
2 + b2
a
√
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
a2
2
r =
+b
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
a = -2
b = -2
r =
in quadrant 3
𝑎2 + 𝑏2 =
= 8= 2 2
Θ = ?
(−𝟐)2 +(−𝟐)2
Using Complex Numbers
z = -2 – 2i
a = -2
b = -2
in quadrant 3
r = 2 2
tanΘ = b/a = -2/-2
So tan-1(1) = Θ
Using Complex Numbers
z = -2 – 2i
a = -2
b = -2
in quadrant 3
r = 2 2
If tan-1(1) = Θ (use your table!)
then Θ = π/4 or 5π/4
Which?
Using Complex Numbers
z = -2 – 2i
a = -2
b = -2
r = 2 2
Θ = 5π/4
in quadrant 3
(in quadrant 3)
Using Complex Numbers
z = -2 – 2i
r = 2 2
Θ = 5π/4
SO what is:
z = r (cos θ +
i
sin θ)
Using Complex Numbers
z = -2 – 2i
r = 2 2
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
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 +
z = 1 +
𝟑
i
i
𝟑/2)
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 +
= -32 + 32
i
i
3/2)
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
a = 1
b = 1
r =
a2
Quadrant 1
+b
2
= ?
DeMoivre’s Theorem
(1 + i)8
a = 1
b = 1
a2
Quadrant 1
2
2
2
r =
+b = 1 +1 =
tan Θ = b/a = ?
2
DeMoivre’s Theorem
(1 + i)8
a = 1
b = 1
a2
Quadrant 1
2
2
2
r =
+b = 1 +1 = 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
a2
2
2
2
r =
+b = 1 +1 =
Θ = π/4 or 45°
1 +
i
2
= 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
= 16(1 + 0 i )
= 16 + 0i or 16
2π)
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!