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Polar Coordinates
In a rectangular coordinate system, we were plotting points based on an ordered pair of (x, y). In the polar coordinate system, the ordered pair will now be (r, θ). The ordered pair specifies a point’s location based on the value of r and the angle, θ, from the polar axis.
The value of r can be positive, negative, or zero. The sign of r is very important in locating the exact position of the point. The absolute value of r, |r|, is the distance between the point and the pole.
- If r is positive (r > 0) then the point lies on the terminal side of θ
- If r is negative (r < 0) then the point lies on the ray opposite of the terminal side of θ
- If r is zero (r = 0) then the point lies at the pole regardless of θ
r > 0 r < 0 r = 0
Example 1: Plot the following polar coordinates.
a) (-3, 135°) b) (2, - 3
π ) c) (4, 405°)
Solution:
One big difference between polar and rectangular coordinates is that polar coordinates can have multiple coordinates representing the same point by adjusting the angle θ or the sign of r and the angle θ.
To find other representations for the point (r, θ):
- Add a multiple of 2π to the angle and do not change r
- Add a multiple of π to the angle and change r to –r
Example 3: Convert the polar coordinates (-8, 2 3
π (^) ) into rectangular coordinates.
Solution:
Plot the polar coordinates
This is not a necessary step but it will provide you with an idea of what should be the rectangular coordinates.
From the graph, we can see that the point is in the fourth quadrant making the x coordinate positive and the y coordinate negative. The rectangular coordinates would be approximately (4, -7).
Find x
x = r cos θ x = -8 cos 2 3
π
x = (-8) (- 1 2
x = 4
Example 3 (Continued):
Find y
y = r sin θ y = -8 sin 2 3
π
y = (-8) ( 3 2
y = -4 3
The rectangular coordinates would be (4, -4 3 ) which is approximately (4, -7).
Example 4: Convert the rectangular coordinates (3, -3) into polar coordinates with r > 0 and 0 ≤ θ < 2π.
Solution:
Plot the rectangular coordinates
From the graph, we can see that the point is in the fourth quadrant.
Example 5: Convert the rectangular equation x^2 + y^2 = 100 into a polar equation that expresses r in terms of θ.
Solution:
Substitute x and y with their polar equivalents
x = r cos θ^ and y = r sin^ θ
x^2 + y^2 = 100 (r cos θ )^2 + (r sin θ ) 2 = 100
Solve for r
r 2 cos^2 θ + r 2 sin^2 θ = 100 r 2 (cos 2 θ + sin^2 θ) = 100 factor common term r^2 r 2 (1) = 100 apply Pythagorean identity cos^2 θ + sin^2 θ = 1 r 2 = 100 r = 10
Example 6: Convert the polar equation 4r cos θ + r sin θ = 8 into a rectangular equation that expresses y in terms of x.
Solution:
Substitute r cos θ and r sin θ with their rectangular equivalents
4r cos θ + r sin θ = 8 4(x) + (y) = 8
Solve for y
4x + y = 8 y = 8 – 4x
