Ch. 5.1 POLAR COORDINATES - A new way of plotting points
In a polar coordinate system, points are represented by ordered
pairs (r, θ) instead of (x,y) in the rectangular coordinate system.
Polar Coordinate System
Rectangular Coordinate System
y-axis
x-axis
P(r, θ)
(x,y)
x
r
y
θ
(0,0)
origin
(0, θ) Polar axis
Pole
Notice the angle can
by any θ at the pole.
2π/3
π/2
What are the polar coordinates of
the following points?
π/3
π/6
5π/6
A = (1, π/6)
B = (3, 5π/3)
π
1
2
3
4
0
C=
D=
7π/6
11π/6
4π/3
5π/3
3π/2
In the polar coordinate system, the r-coordinates are all considered positive if they lie on the
terminal side of the ray with angle θ. The r-coordinate is considered negative if it is on the ray
from the pole extending in the direction opposite the terminal side of θ at a distance |r| from the
pole.
In the polar coordinate system, points may be expressed in multiple ways. Point A can be expressed
as (1, π/6) or also as (-1, 7π/6), or also as (1, 13π/6) or (-1, 19π/6), etc…
Therefore, a point with polar coordinates (r, θ) can also be represented by:
(r, θ+2kπ) or (-r, θ+π+2kπ) for any integer k.
What are some alternative polar coordinates for points B, C and D?
POLAR COORDINATE SYSTEM
π/2 = -3π/2
90°=-270°
2π/3 =- 4π/3
120°=-240°
π/3 =- 5π/3
60°= -300°
3π/4= -5π/4
135°=- 225°
π/4 =- 7π/4
45°=-315°
5π/6= - 7π/6
150°=-210 °
π/6 = -11π/6
30°=-330°
π= - π
180°=- 180°
1
2
3
4
7π/6=- 5π/6
210°=- 150°
5π/4= -3π/4
225°=- 135°
11π/6= - π/6
330°= - 30°
7π/4= - π/4
315°= - 45°
4π/3=- 2π/3
240°= - 120°
5π/3 = -π/3
300°=- 60°
3π/2=- π/2
270°=- 90°
Do #21
0 =2π = -2π
0° = 360°
y
x
CONVERTING FROM POLAR COORDINATES TO
RECTANGULAR COORDINATES
Notice that x/r = cos θ, and y/r = sin θ , so solving for x and y gives
x = r cos θ
y = r sin θ
Example 5
Find the rectangular coordinates of the following points:
a)
b)
⎛ π⎞
⎜ 6, ⎟
⎝ 6⎠
π⎞
⎛
⎜ − 4,− ⎟
4⎠
⎝
⎛ 3⎞
⎟=3 3
x = 6 cos = 6⎜⎜
⎟
2
6
⎝
⎠
π
⎛1⎞
y = 6 sin = 6⎜ ⎟ = 3
6
⎝2⎠
⎛ 2⎞
⎛ π⎞
⎟ = −2 2
x = −4 cos⎜ − ⎟ = −4⎜⎜
⎟
⎝ 4⎠
⎝ 2 ⎠
⎛
2⎞
⎛ π⎞
⎟=2 2
y = −4 sin ⎜ − ⎟ = −4⎜⎜ −
⎟
4
2
⎝
⎠
⎠
⎝
( x, y ) = 3 3 ,3
( x, y ) = − 2 2 , 2 2
π
(
)
(
)
Now
Do
#29
And
#41
CONVERTING FROM RECTANGULAR COORDINATES
TO POLAR COORDINATES
Given (x,y), find (r, θ)
From the Pythagorean Theorem we get:
P(r, θ) x2 + y2 = r2
r
θ
x
y
θ can be found using that fact that tan θ = y/x
so θ = tan-1(y/x)
However, this solution only works for –π/2 <θ < π/2.
Knowing which Quadrant (x,y) is in will tell you what you have to add
to your result of tan-1(y/x).
Steps for Converting from Rectangular tro Polar Coordnates
Step 1: Determine which Quadrant (x,y) is in.
Step 2: To find r, use x2 + y2 = r2
Step 3: To find θ, use the following rules for (x,y) being in each
particular quadrant.
Quadrant I : θ = tan-1(y/x)
Quadrant II: θ = π + tan-1(y/x)
Quadrant III: θ = π + tan-1(y/x)
Quadrant IV:θ = tan-1(y/x)
What about if x = 0?
If x = 0 the inverse tangent of y/x is undefined. However, let’s look at
the point (0,a). For a > 0, the point will be at θ = π/2 and r = a.
For a< 0, the point will be at r = a and θ = π/2,or alternatively,
r = -a and θ = -π/2 (or θ = 3π/2, etc..)
Rectangular Coordinate System
(0,a)
y-axis
x-axis
(0,0)
origin
HOMEWORK
p. 326 # 1,7, 15, 17, 21, 23, 29, 35, 47, 49