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Chapter 11: Polar Coordinates and Complex Numbers

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Chapter 11: Polar Coordinates and Complex Numbers

L11.1 Polar Coordinates and Graphs

Polar Coordinates offer another way to locate a point on a plane. Applications: radar, GPS, ...

Rectangular Coord System:

Polar Coordinate System has:

- a point, O, called the pole
- an initial ray from O called the polar axis
Each point, P, in the plane can be assigned a polor coord (r, θ)

Lesson 11.1:

- Plotting points in the Polar Coordinate System
- Coordinate conversion (Polar ↔ Rectangular)
- Sketching Polar Graphs (Graphing equations, recognizing patterns)
- Equation conversion (Polar ↔ Rectangular)

~ origin

~ positive x-axis

P

r is directed distance from O (pole)

r

θ is directed angle from polar axis

θ

pole

polar axis

The point (2, π/3) lies at the intersection of a circle with radius 2, and the terminal side of the angle π/3:

Plot these: 1. (1, 3π/4), 2. (3, 210°)

Unlike the rectangular coordinate system, a point in the polar coordinate system can have multiple representations:

(r, θ) = (r, θ ± 2πn), n is an integer

Thus, (2, π/3) = (2, 7π/3) = (2, −5π/3) = …

If r < 0, find the ray that forms the angle with the polar axis and go in the opposite direction from the ray. For ex, (−2, π/3):

Thus (−2, π/3) = (2, 4π/3)

(−r, θ) = (r, θ ± π)

y

y

P(x, y)

P(x, y)

r

r

y

y

ɵ

O

O

x

x

x

x

Recall the circular definitions from Chapter 7:

Convert from polar to rectangular:

Ex. Convert (2, π) to rectangular coordinates.

Convert from rectangular to polar:

Ex: Convert (−1 , 1) to polar coordinates.

x2 + y2 = r2

x = 2 cos π = −2, y = 2 sin π = 0, thus (−2, 0)

On Ti89:

MATH → Angle

P►Rx(

P►Ry(

R►Pr(

R►Pθ(

Thus,

Polar→Rect

tan θ = 1/−1= −1, θ is in Q2, so θ = 3π/4.

Rect→Polar

- Plot these points, write in another polar form and convert to rectangular form:
- (3, π/4)
- (−⅔, 7π/6)

- Convert to polar coordinate form, and provide 2 additional equivalent polar representations:
- (−4, 3)
- (2, −4)

(r, θ) = (r, θ ± 2πn), n is an integer

(−r, θ) = (r, θ ± π)

Ready for answers?

- To graph an equation in the rectangular system, plot y in terms of x; i.e., y = f(x).
- Create an x/y table, fixing x values and computing corresponding y values. Then connect the dots.
- You also use knowledge of the equation to guide the graph.

- When you graph an equation in the polar system, you plot r in terms of θ; i.e., r = f(θ).
- Create an θ/r table, fixing θ values and computing corresponding r values. Then connect the dots.
- You also use knowledge of the equation to guide the graph.

- Example: r = 4 sin θ

0

−2

2

−4

4

−2

0

2

In this case, 0→π completed the graph; π →2π retraced it.

0

- You can also use knowledge of the eqn to guide the graph:
- Use Symmetry
- r = f(sin θ) is symm wrt the line θ = π/2
- r = f(cosθ) is symm wrt the polar axis

- Determine θ value(s) that produce max r values
- Determine θ value(s) that produce zero r values

- Use Symmetry
- Previous example: r = 4 sin θ
- Symm wrt the line θ = π/2
- Max when sin θ = 1, i.e., when θ = π/2; value is 4
- Zero when sin θ = 0, i.e., when θ = 0 or θ = π

- Example: r = 1 – 2 cosθ
- Symm wrt the polar axis
- Max when cos θ = −1 or θ = π; value is 3
- Zero when cosθ = ½ or θ = π/3

−1

−.73

0

1

2

2.73

3

Plot points and connect; Then use symmetry to complete figure.

→Limacon with inner loop.

r = aθ

Lemiscates

Circles

a

a

Archimedan Spiral

Limacons: r = a ± b cos θ, r = a ± b sin θ, a > 0, b > 0

a

a

a

r = a

r2 = a2 cos 2θ

r2 = a2 sin 2θ

r = a cos θ

r = a sin θ

Cartoid: a/b = 1

Limacon w/ inner

loop: a/b < 1

Dimpled Limacon:

1 < a/b < 2

Convex Limacon:

a/b > 2

Rose Curves: n petals if n is odd; 2n petals if n is even (n ≥ 2)

n = 5

n = 3

n = 6

n = 4

a : length of petals

r = a sin nθ

r = a cos nθ

Graphing mode = polar. Must also be in radian mode.

Let’s graph r = 5 sin 4θ.

Press to draw the graph.

Press to draw the graph.

Press & changethe graph mode to

3:POLAR. Press

Enter to Save.

Enter Y=Editor. Press forthe variable, θ.

Redraw using

ZoomSqr.

Set ZoomTrig.

Confirm window by

pressing

- Sometimes a very simple polar equation is complex when converted to rectangular form (or vice-versa)
- Conversion factors:
- x = r cos θ y = r sin θ x2 + y2 = r2
- Rectangular to Polar y = f(x) → r = f(θ) or y2 → r2
- x2 + y2 = 9 2. 3. x2 + y2 – 2ay = 0

- Polar to Rectangular r = f(θ) → y = f(x) or r2 → y2
- r = 2 2. 3. r = sec θ
- r = 4 sin θ 5. r = 2 cscθ