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. 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

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Chapter 11: Polar Coordinates and Complex Numbers

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Chapter 11 polar coordinates and complex numbers

Chapter 11: Polar Coordinates and Complex Numbers

L11.1 Polar Coordinates and Graphs


Polar coordinates

POLAR COORDINATES

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


I plotting points in the polar coordinate system

I. Plotting points in the Polar Coordinate System

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)


Ii coordinate conversion polar rectangular

II. Coordinate conversion (Polar ↔ Rectangular)

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


I ii polar coordinates examples

I & II. Polar Coordinates: Examples

  • 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?


Iii sketching polar graphs

III. Sketching Polar Graphs

  • 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


Iii sketching polar graphs1

III. Sketching Polar Graphs

  • 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

  • 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


Iii sketching polar graphs2

III. Sketching Polar Graphs

  • 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.


Simple polar equations can produce beautiful graphs

Simple polar equations can produce beautiful graphs

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


Iii sketching polar graphs ti89

III. Sketching Polar Graphs: Ti89

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


Iv equation conversion polar rectangular

IV. Equation conversion (Polar ↔ Rectangular)

  • 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


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