1. Polar Coordinates Polar and Rectangular Coordinates
9.3

2. Polar vs. Rectangular Coordinates
For some real-world phenomena, it is useful to be able to convert between polar coordinates and rectangular coordinates.

3. Polar vs. Rectangular Coordinates
Polar Coordinates: P(r, θ)
Rectangular Coordinates: P(x, y)
Suppose a rectangular coordinate system is superimposed on a polar coordinate system so that the origins coincide and the x-axis aligns with the polar axis, as shown at the right. Let P be any point in the plane.

4. Polar to Rectangular Coordinates
Trigonometric functions can be used to convert polar coordinates to rectangular coordinates.
The rectangular coordinates (x, y) of a point named by the polar coordinates (r, θ) can be found by using the following formulas:
x = r cos θ
y = r sin θ

5. Rectangular to Polar Coordinates
If a point is named by the rectangular coordinates (x, y), you can find the corresponding polar coordinates by using the Pythagorean Theorem and the Arctangent function (Arctangent is also known as the inverse tangent function).

6. Rectangular to Polar Coordinates
Since the Arctangent function only determines angles in quadrants 1 and 4 (because tangent has an inverse in quadrants 1 and 4) you must add π radians to the value of θ for points with coordinates (x, y) that are in quadrants 2 or 3.

7. Rectangular to Polar Coordinates
When x > 0, θ =
When x < 0, θ =

8. Rectangular to Polar Coordinates
When x is zero, . Why?
The polar coordinates (r, θ) of a point named by the rectangular coordinates (x, y) can be found by the following formulas:
r =
When x > 0
When x < 0

9. Converting Equations
The conversion equations can also be used to convert equations from one coordinate system to the other.

10. Example 1 Find the rectangular coordinates of each point.
a b. C(3, 270°)

11. Example 2
Find the polar coordinates of
a) E(2, -4) b) F(-8, -12)

12. Example 3 Write the polar equation r = -3 in rectangular form.
Write the polar equation r = 5 cosθ in rectangular form.

13. Example 4
Write the rectangular equation x² + (y – 1)² = 1 in polar form.