8.5 Polar CoordinatesThe 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, )
0 degrees
90 degrees
180 degrees
270 degrees
r
Examples to try (3, 30°) (2, 135) (-2, 30°) (-1, -45)
Converting Coordinates: Polar to/from Rectangular
0 degrees
90 degrees
180 degreesx
y
P(x,y)r
x = r cos y = r sin
x2 + y2 = r2
tan = y x
Note: You can also convert rectangularEquations to polar equations and vice versa.
Example: Polar to RectangularPolar Point: (2, 30º)X = 2 cos 30 = 3Y = 2 sin 30 = 1Rectangular point: (3 , 1)
Example: Rectangular to PolarRextangular Point: (3, 5)32 + 52 = r2 r = 34
tan = 5/3 = 59ºPolar point: (34 , 59º)
Rectangular vs Polar EquationsRectangular equations are written in x and yPolar equations are written with variables r and
Rectangular equations can be written in an equivalent polar form
x = r cos y = r sin
x2 + y2 = r2
tan = y 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 EquationsRectangular equations are written in x and yPolar equations are written with variables r and
Polar equations can be written in an equivalent rectangular form
x = r cos y = r sin
x2 + y2 = r2
tan = y x
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 EquationsTo Graph a polar equation,Make a / r chart for until a pattern apppears.Then join the points with a smooth curve.
r
Example: r = 3 cos 2 (4 leaved rose)
0153045607590
32.61.50-1.5-2.6-3
0 degrees
90 degrees
180 degrees
270 degrees
P. 387 in your text shows various types of polar graphs and associated equation forms.
Graphing Polar EquationsTo Graph a polar equation,Make a / r chart for until a pattern apppears.Then join the points with a smooth curve.
r
Example: r = 3 cos 2 (4 leaved rose)
0153045607590
32.61.50-1.5-2.6-3
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
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 t/x/y chart, then graphing points (x,y)T x y-3 9 -3-2 4 -1-1 1 10 0 31 1 52 4 73 9 9
Convert to rectangular form byEliminating the parameter ‘t’
Step 1: Solve 1 equation for tStep 2: Substitute ‘t’ into the ‘other’ equation
Y = 2t + 3 t = (y – 3)/2
X = ((y – 3)/2)2
X = (y – 3)2
4
Parametric Equations are sometimes used to simulate ‘motion’
•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.
Application: Toy Rocket
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, thenProduct 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 angleIs 45 degrees so θ = 315 degrees.
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 Example from your book
Product Rule
•Find the quotient
Quotient Rule Example from your Book
Note: CIS 45◦ is an abbreviationFor (cos 45◦+ isin 45◦)
Quotient Rule
8.4 De Moivre’s Theorem
is a complex number, then
Example: Find (1 + i3)8 and express the result in rectangular form1st, express in Trig Form: 1 + i3 = 2(cos 60 + i sin 60)Now apply De Moivre’s Theorem:
480° and 120° are coterminal.
Rectangular form