Transcript 8.5 Polar Coordinates

8.5 Polar Coordinates
The rectangular coordinate system (x/y axis) works in 2 dimensions with each
point having exactly one representation.
A polar coordinate system allows for the rotation and repetition of points.
Each point has infinitely many representations.
A polar coordinate point is represented by an ordered pair
(r, )
90 degrees
r
180 degrees

270 degrees
0 degrees
Examples to try
(3, 30°)
(2, 135)
(-2, 30°)
(-1, -45)
Converting Coordinates:
Polar to/from Rectangular
90 degrees
P(x,y)
r

180 degrees
x
Example: Polar to Rectangular
Polar Point: (2, 30º)
X = 2 cos 30 = 3
Y = 2 sin 30 = 1
Rectangular point: (3 , 1)
y
0 degrees
Example: Rectangular to Polar
Rextangular Point: (3, 5)
32 + 52 = r2  r = 34
x = r cos 
tan  = 5/3   = 59º
Polar point: (34 , 59º)
y = r sin 
x2 + y2 = r2
Note: You can also convert rectangular
Equations to polar equations and vice versa.
tan  = y
x
Rectangular vs Polar Equations
Rectangular equations are written in x and y
Polar equations are written with variables r and 
x = r cos 
y = r sin 
x2 + y2 = r2
tan  = y
Rectangular equations can be written in an equivalent polar form
x
Example1: Convert y = x - 3 (equation of a line) to polar form.
 x–y=3
 (r cos ) – (r sin ) = 3
 r (cos  - sin ) = 3
 r = 3/(cos  - sin )
Example2: Convert x2 + y2 = 4 (equation of circle) to polar form
 r2 = 4
 r = 2 or r = -2
Rectangular vs Polar Equations
Rectangular equations are written in x and y
Polar equations are written with variables r and 
x = r cos 
y = r sin 
x2 + y2 = r2
tan  = y
x
Polar equations can be written in an equivalent rectangular form
Example1: Convert
to rectangular form.
 r + rsinθ = 4
r+y=4=3


 x2 + y2 = (4 – y)2
 x2 = -y2 + 16 -8y + y2
x2 = 16 -8y
x2 – 16 = -8y
y = - (1/8) x2 + 2
Graphing Polar Equations
To Graph a polar equation,
Make a  / r chart for until a pattern apppears.
Then join the points with a smooth curve.
Example: r = 3 cos 2 (4 leaved rose)
90 degrees

r
0
15
30
45
60
75
90
3
2.6
1.5
0
-1.5
-2.6
-3
180 degrees
0 degrees
270 degrees
P. 387 in your text shows various types of polar graphs and associated equation forms.
Graphing Polar Equations
To Graph a polar equation,
Make a  / r chart for until a pattern apppears.
Then join the points with a smooth curve.
Example: r = 3 cos 2 (4 leaved rose)

r
0
15
30
45
60
75
90
3
2.6
1.5
0
-1.5
-2.6
-3
P. 387 in your text shows various types of polar graphs and associated equation forms.
Classifying Polar Equations
• Circles and Lemniscates
• Limaçons
• Rose Curves 2n leaves if n is even n ≥ 2 and n leaves if n is odd
8.6 Parametric Equations
Parametric Equations are sometimes used to simulate ‘motion’
x = f(x) and y = g(t) are parametric equations with parameter, t when they
Define a plane curve with a set of points (x, y) on an interval I.
Example: Let x = t2 and y = 2t + 3 for t in the interval [-3, 3]
Graph these equations by making a
Convert to rectangular form by
t/x/y chart, then graphing points (x,y) Eliminating the parameter ‘t’
T x y
Step 1: Solve 1 equation for t
-3 9 -3
Step 2: Substitute ‘t’ into the
-2 4 -1
‘other’ equation
-1 1 1
0 0 3
Y = 2t + 3  t = (y – 3)/2
1 1 5
2 4 7
2
X
=
((y
–
3)/2)
3 9 9
X = (y – 3)2
4
Application: Toy Rocket
•A toy rocket is launched from the ground with velocity 36 feet per second at
an angle of 45° with the ground. Find the rectangular equation that models
this path. What type of path does the rocket follow?
The motion of a projectile (neglecting air resistance) can be modeled by
for t in [0, k].
Since the rocket is launched from the ground, h = 0.
The parametric equations determined by the toy rocket are
Substitute from Equation 1 into equation 2:
A Parabolic Path
8.2 & 8.3 Complex Numbers
Graphing Complex Numbers:
• Use x-axis as ‘real’ part
• Use y-axis as ‘imaginary’ part
Trig/Polar Form of Complex Numbers:
• Rectangular form: a + bi
• Polar form: r (cos  + isin )
are any two complex numbers, then
Product Rule
Quotient Rule
Examples of Polar Form Complex
Numbers
Trig/Polar Form of Complex Numbers:
• Rectangular form: a + bi
• Polar form: r (cos  + isin )
Example 1:
Express 10(cos 135° + i sin 135°) in
rectangular form.
Example2:
Write 8 – 8i in trigonometric form.
The reference angle
Is 45 degrees so θ = 315 degrees.
Product Rule Example from your book
Find the product of 4(cos 120° + i sin 120°) and 5(cos 30° + i sin 30°).
Write the result in rectangular form.
Product Rule
Quotient Rule Example from your Book
•Find the quotient
Note: CIS 45◦ is an abbreviation
For (cos 45◦+ isin 45◦)
Quotient Rule
8.4 De Moivre’s Theorem
is a complex number, then
Example: Find (1 + i3)8 and express the result in rectangular form
1st, express in Trig Form: 1 + i3 = 2(cos 60 + i sin 60)
Now apply De Moivre’s Theorem:
480° and 120° are coterminal.
Rectangular form