• P. 783: 1-8 S P. 783: 9-16 S P. 783: 17-26 S

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P. 783: 1-8 S
P. 783: 9-16 S
P. 783: 17-26 S
P. 783: 33-48 S
P. 784: 49-64 S
Start at the origin
facing due east (the
positive 𝑥-axis). Now
rotate π/3 radians,
and then walk 5 units.
What are the exact
coordinates of your
new location?
Objectives:
1. To graph points in
polar coordinates
2. To convert between
rectangular and
polar coordinate
systems
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Assignment:
P. 783: 1-8 S
P. 783: 9-16 S
P. 783: 17-26 S
P. 783: 33-48 S
P. 784: 49-64 S
In a rectangular
coordinate system,
you locate points as
directed distances
from the 𝑥- and 𝑦axes.
Pole
In a polar coordinate
system, you locate
points with an angle
of rotation 𝜃 from
one axis called the
polar axis, and a
directed distance 𝑟
from the origin or
pole.
Polar Axis
1. 𝑟 = directed
distance from 𝑂 to 𝑃
– Think of this as the
𝑟𝑎𝑑𝑖𝑢𝑠 of a circle
Pole
2. 𝜃 = directed angle,
counterclockwise
from polar axis to
𝑂𝑃
Polar Axis
Plot the following points in polar coordinates.
1.
𝜋
3, 3
2.
𝜋
−3, 3
Plot the following points in polar coordinates.
3.
𝜋
3, − 3
4.
𝜋
−3, − 3
Positive
Negative
𝒓
Directed
distance is in the
same direction
as the angle
Directed
distance is in the
exact opposite
direction from
the angle
(180°or 𝜋 rad)
𝜽
Counterclockwise
rotation from
polar axis
Clockwise
rotation from
polar axis
Plot the following points in polar coordinates.
1.
𝜋
2, 6
2.
𝜋
2, − 6
3.
𝜋
−2, 6
4.
𝜋
−2, − 6
If you plot a point in
the rectangular
coordinate system,
there’s only one
ordered pair that
represents that
point. It is unique.
In polar coordinates, though, any point can have
multiple representations.
5, 𝜋3 → 5, 7𝜋
3
↓
−5, 4𝜋
3
Original
Point
Point In The Same
Location
𝑟, 𝜃
𝑟, 𝜃 + 2𝑛𝜋
𝑟, 𝜃
−𝑟, 𝜃 + 2𝑛 + 1 𝜋
In polar coordinates, though, any point can have
multiple representations.
Another example:
𝜋 2
0,0 ↔ 0, −𝜋 ↔ 0, 𝜋7 ↔ 0, −3𝜋
↔
0,
2
5
Plot the point 2, 3𝜋
and find 3 additional polar
4
representations of this point within the interval
− 2𝜋 < 𝜃 < 2𝜋.
Plot the rectangular point 4, 3 . Now convert
this point to polar coordinates.
To convert rectangular coordinates to polar
coordinates:
𝑦
tan 𝜃 =
𝑥
𝑟2 = 𝑥2 + 𝑦2
𝑟, 𝜃
Plot the polar point 5, 𝜋3 . Now convert this
point to rectangular coordinates.
To convert polar coordinates to rectangular
coordinates:
𝑥 = 𝑟 ∙ cos 𝜃
𝑦 = 𝑟 ∙ sin 𝜃
𝑥, 𝑦
Convert the following rectangular coordinates to
polar coordinates.
1.
2, 2
2.
−1, 0
3.
6, 7
Convert the following polar coordinates to
rectangular coordinates.
1.
4, −3𝜋
2
2.
2, 𝜋6
3.
−5, 𝜋5
To convert an equation
in the rectangular
coordinate plane to
the polar plane,
substitute:
𝑥 = 𝑟 ∙ cos 𝜃
𝑦 = 𝑟 ∙ sin 𝜃
𝑦 = 𝑥2
𝑟 ∙ sin 𝜃 = 𝑟 ∙ cos 𝜃
2
𝑟 ∙ sin 𝜃 = 𝑟 2 ∙ cos 2 𝜃
𝑟 ∙ sin 𝜃 𝑟 2 ∙ cos 2 𝜃
=
𝑟
𝑟
sin 𝜃 = 𝑟 ∙ cos 2 𝜃
sin 𝜃
𝑟 ∙ cos 2 𝜃
=
cos2 𝜃
cos 2 𝜃
sin 𝜃
1
∙
=𝑟
cos 𝜃 cos 𝜃
𝑟 = tan 𝜃 sec 𝜃
Convert 𝑦 = 2𝑥 + 1 to a polar equation.
Convert 𝑥 2 + 𝑦 2 = 25 to a polar equation.
To convert an equation
in polar coordinate
plane to the
rectangular plane,
use:
tan 𝜃 = 𝑦𝑥
𝑟2 = 𝑥2 + 𝑦2
𝑟=2
2
2
= 𝑥2 + 𝑦2
4 = 𝑥2 + 𝑦2
Circle with radius = 4
To convert an equation
in polar coordinate
plane to the
rectangular plane,
use:
𝜃 = 𝜋3
tan
𝜋
3
= 𝑦𝑥
3 = 𝑦𝑥
𝑥 3=𝑦
Line with slope = 3
tan 𝜃 = 𝑦𝑥
𝑟2 = 𝑥2 + 𝑦2
To convert an equation
in polar coordinate
plane to the
rectangular plane,
use:
𝑟 = sec 𝜃
𝑥
cos 𝜃
= sec 𝜃
cos 𝜃 ∙ cos𝑥 𝜃 = sec 𝜃 ∙ cos 𝜃
𝑥 = cos1 𝜃 ∙ cos 𝜃
𝑥=1
tan 𝜃 = 𝑦𝑥
𝑟2 = 𝑥2 + 𝑦2
Vertical line through 1
Convert 𝑟 = 1 to a rectangular equation.
Convert 𝜃 = 𝜋4 to a rectangular equation.
Convert 𝑟 = 2sin 𝜃 to a rectangular equation.
Objectives:
1. To graph points in
polar coordinates
2. To convert between
rectangular and
polar coordinate
systems
•
•
•
•
•
Assignment:
P. 783: 1-8 S
P. 783: 9-16 S
P. 783: 17-26 S
P. 783: 33-48 S
P. 784: 49-64 S