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