9.6 Polar Coordinates Digital Lesson. HWQ 3/24 Find a set of parametric equations to represent the...

9.6 Polar Coordinates

Digital Lesson

One way to give someone directions is to tell them to go three blocks East and five blocks South.

Another way to give directions is to point and say “Go a half mile in that direction.”

Polar graphing is like the second method of giving directions. Each point is determined by a distance and an angle.

Initial ray

r A polar coordinate pair

determines the location of a point.

,r

9.6 Polar Coordinates

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4

The polar coordinate system is formed by fixing a point, O, which is the pole (or origin).

= directed angle Polar axis

r = directed dista

nce

OPole (Origin)

The polar axis is the ray constructed from O.

Each point P in the plane can be assigned polar coordinates (r, ).

P = (r, )

r is the directed distance from O to P.

is the directed angle (counterclockwise) from the polar axis to OP.


a) The point lies two units from the pole on the

terminal side of the angle

( , ) 2,3

r

.3

3

2,3

2

32

1 2 3 0

Plotting Points


33,4

34

2

32

1 2 3 0

3 units from the pole

Plotting Points

b) Plot the point 33,

4


There are many ways to represent the point 2, .3

2

32

1 2 3 0

2,3

52,3

2 42,2 2,,3 3 3

52,3

additional ways

to represent the

point 2,3

( , ) , 2r r n

( , ) , (2 1)r r n

• Plot the polar point , then find one additional representation.


3,4

2

32

1 2 3 0


(r, )(x, y)

Polex

y

(Origin)

y

r

x

The relationship between rectangular and polar coordinates is as follows.

The point (x, y) lies on a circle of radius r, therefore,

r2 = x2 + y2.

tan yx

cos xr

sin yr

Definitions of trigonometric functions


Coordinate Conversion

cosx r cos xr

siny r sin yr

2 2 2r x y tan yx

(Pythagorean Identity)

Example:Convert the point into rectangular coordinates. 4,

3

1cos co3

24 s 42

x r

3sin sin 4 23 2

4 3y r

, 2, 2 3x y

x = r cos(θ)

y = r sin(θ)

r2 = x2 + y2

tan(θ) = y/x


Example:

Convert the point into rectangular coordinates. 2,

Example:

Convert the point into rectangular coordinates. 3,6

Example:

Convert the point into rectangular coordinates. 4.5,1.3

x = r cos(θ)

y = r sin(θ)

2,0

3 3,

2 2

1.204,. 4.336


Example:

Convert the point (1,1) into polar coordinates.

, 1,1x y

1tan 11

yx

4

2 2 2 21 1 2r x y

set of polar coordinates is ( , ) 2, .4

One r

5Another set is ( , ) 2, .4

r


Example:

Convert the point (0,2) into polar coordinates.

Example:

Convert the point (-3,4) into polar coordinates.

r2 = x2 + y2

tan(θ) = y/x

2,2

5, 2.214


Equation Conversion

x = r cos(θ)

y = r sin(θ)

r2 = x2 + y2

tan(θ) = y/x

To convert a rectangular equation to polar form, use:

To convert a polar equation to rectangular form, use:

Equation Conversion

Example For the rectangular equation 3x + 2y = 4,

(a) convert to a polar equation,

(b) use a graphing calculator to graph the rectangular equation, and

(c) use a graphing calculator to graph the polar equation for 0° 360°.

Solution

(a) Let x = r cos and y = r sin to get

3 cos 2 sin 4 or4

.3cos 2sin

rr r

Converting a Rectangular Equation to Polar Form

(b) Solve the rectangular equation for y to get

(c)

.223

xy

Converting a Polar Equation to Rectangular Form

Example For the polar equation

(a) convert to a rectangular equation,

(b) use a graphing calculator to graph the polar equation for 0 2, and

(c) use a graphing calculator to graph the rectangular equation.

Solution: Multiply both sides by the denominator.

41 sin ,r

sin 4r r 2 2 4x y y 2

22 2 4x y y

2 2 216 8x y y y 2 8 2x y

Converting a Polar Equation to Rectangular Form

(b) The figure showsa graph with polarcoordinates.

(c) Solving x2 = –8(y – 2)for y, we obtain

.2 281 xy

Homework Day 1

9.6 pg. 680:

1, 5, 9, 13, 17, 21, 25, 29, 33, 37-55 odds


9.6 Polar CoordinatesDay 2

Digital Lesson

Answer equation conversion problems at end of lesson.

HWQ 3/25

Find 2 different sets of polar coordinates

for the rectangular point:

Exact values only.


3, 1


9.6 Day 2 - More Equation Conversion

x = r cos(θ)

y = r sin(θ)

r2 = x2 + y2

tan(θ) = y/x

To convert a rectangular equation to polar form, use:

To convert a polar equation to rectangular form, use:


Converting Equations

Ex: Convert the rectangular equation to polar form: 2y x

sec tanr


Ex: Convert the rectangular equation to polar form: 2 2 16x y

4r


3 6 2 0x y Ex: Convert the rectangular equation to

polar form:

2

6sin 3cosr


Equation Conversion

Convert the polar equation into a rectangular equation.

4sinr

4sinr 2 4 sinr r Multiply each side by r.

2 2 4x y y Substitute rectangular coordinates.

22 2 4x y Equation of a circle with center (0, 2) and radius of 2

Polar form

2 2 4 0x y y


Example:

Convert the polar equation r = 2 to rectangular form.

2 2 4x y


Example:

Convert the polar equation to rectangular form.

3

3y x

tan tan3

3y

x


Example:

Convert the polar equation to rectangular form. secr

x = r cos(θ)

y = r sin(θ)

cos sec cosr

1x sec cosx


Example:

Convert from rectangular to polar form:2 3x y

2cot cscr


Example:


x = r cos(θ)

y = r sin(θ)

3cosr

223 9

2 4x y


Example:


x = r cos(θ)

y = r sin(θ)

2 sin 2r

22 2 2x y xy

Homework Day 2

9.6 pg. 680

3, 7, 11, 15, 19, 23, 27, 31, 35, 57-77 odds


HWQ 3/26

• Convert the rectangular equation to polar form:


2 2 8 0x y y

9.6 Polar Coordinates Digital Lesson. HWQ 3/24 Find a set of parametric equations to represent the...

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