We can look at rectangular and polar coordinates as two different languages. We use them to describe the same things using different words. In the last section we learned how to translate points from one coordinate system to another.
Begin with a basic example, the right triangle. Rectangular and polar coordinates give different information about a right triangle.
From looking at the triangle we can see that these statements are true:
These transformation equations let us translate equations and inequalities between different coordinates.
To translate an equation or inequality from rectangular to polar coordinates, x becomes r cos θ and y becomes r sin θ. We can also replace x2 + y2 with r2.
Translating from polar to rectangular coordinates isn't quite as straightforward as going the other way. We can replace r2 with x2 + y2, r cos θ with x, and r sin θ with y. However, sometimes we might need to do an extra step or two before we have any recognizable terms to replace.
Why can't we use one coordinate system and be done with it?
The answer is that some equations and inequalities look better in one system than the other. We know from practice that the equation
x2 + y2 = 36
describes a pizza with a 6 inch radius, but isn't
r = 6
a much nicer way to describe it?
Going the other way,
the polar inequality
r ≤ 2/cos θ
is a mess. If we translate into rectangular coordinates, though, we find
r cos θ ≤ 2
x ≤ 2
which describes the part of the (x, y)-plane that lies on and to the left of the line x = 2.
If we're good at translating between coordinate systems, we can quickly find the simplest representation of a particular equation.