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It's a hot summer evening. Bob has been training for a triathalon and needs a major calorie intake. He finished his dinner of pizza and hot dogs, but he still has room for dessert, so he eyeballs that key lime pie his neighbor brought over for him yesterday.
He walks over to the counter, licks his lips, and picks up a knife. He drives the knife right into the center of the pie and makes a cut. He turns the pie, places the knife in the center again, and cuts himself a heathy-sized wedge.
Polar coordinates are the first choice for describing circular things, key lime pie notwithstanding.
We often describe the location of a point by giving its rectangular or Cartesian coordinates (x, y). The first coordinate x describes how far we travel to the right (or left) of the origin, and the second coordinate y describes how far we travel up (or down).
We can also describe the position of a point using polar coordinates of the form (r, θ). Polar coordinates are closely related to vectors. Instead of listing the x and y components of a vector we list its magnitude |r| and its direction θ.
To graph the point whose polar coordinates are (r, θ), first draw a ray from the origin with angle θ from the positive x-axis:
If r is positive, start at the origin and travel a distance of r along the ray:
If r is negative, start at the origin and travel a distance of |r| in the opposite direction:
If we know the angle θ and whether r is positive or negative, we know which quadrant the point (r, θ) can be found.
In order to make polar coordinates more digestable, make sure we're comfortable thinking about angles in radians. The next time someone asks how much pie you want, tell them you want radians. It would also be helpful to review the unit circle, since polar coordinates are sort of an extension of the unit circle. It's no coincidence, seeing as pizzas and pies are shaped the same way.
Be Careful: With polar coordinates there are infinitely many ways to describe each point; it can loop around forever. With rectangular coordinates there's only one way to describe each point.
The point can be described by the polar coordinates .
It can also be described by the coordinates for any integer value of n. Every time we add 2π, we go around the circle and end up where we started:
We usually restrict our polar coordinates such that 0 ≤ θ < 2π and r is positive. Then there will be only one way to describe each point in polar coordinates.
Say we want a piece of a pizza, but we don't want the crust. The crust gets stuck in between our teeth, and we have a dentist appointment right after lunch.
The pizza has a 6 inch radius with the crust, but the crust is a half inch thick. When we order our slice, we ask for a slice with only the inner 5.5 inch radius. Naturally, we receive an odd look from the guy behind the counter, but are given your pizza sans crust.
Like crustless pizza, there are certain situations that are easier to describe with polar inequalities than with rectangular inequalities. Sometimes giving bounds for r and θ is easier than giving bounds for x and y.
Of course we can also go the other way around, starting with inequalities and ending up with a picture.
We've really enjoyed our slice of pie, and we've even gone back for seconds while nobody was watching. We need to pack the rest into a box and take it with us to share, though. We can only find square-shaped boxes, so how do we know if the last 2⁄3 of the pie is going to fit into a box?
We would need to have a way to translate from polar to rectangular coordinates and vice versa. This is no problem, since we can describe where a point is using either polar or rectangular coordinates, and we only need a few tools to switch between coordinate systems.
We'll start with points in the first quadrant. Any point in the first quadrant can be described by rectangular coordinates
(x, y) where x, y ≥ 0 or by polar coordinates
(r, θ) where r ≥ 0 and
To translate between coordinate systems, draw a right triangle whose hypotenuse connects the origin and the point. One leg of the triangle should be on the x-axis.
The rectangular coordinates tell us the lengths of the legs of the triangle:
The polar coordinates tell us the hypotenuse and one angle of the triangle (since it's a right triangle, we know all the angles):
Each set of coordinates is telling us different information about this triangle. We can use one set of coordinates and a small selection of our tools for working with triangles to find the other set of coordinates.
Going from rectangular to polar coordinates is the same thing as finding the magnitude and direction of a vector.
To go from polar to rectangular coordinates, remember the definitions of the sine and cosine functions.
It's a little trickier to translate between coordinates when the points are in other quadrants.
The first thing we do, no matter which way we're translating, is recognize which quadrant the point is in. After translating between coordinates, the point should still be in the same quadrant. This will help us know if our answer is reasonable.
To go from rectangular to polar coordinates is still the same thing as finding the magnitude and direction of a vector. To find the direction we may need to use a reference angle.
To go from polar to rectangular coordinates, we don't need to worry about reference angles. All we need to do is use the definitions of sine and cosine.
Translating points between rectangular and polar coordinates may be a bit tedious. Going from rectangular to polar coordinates is the same thing as
finding the magnitude and direction of a vector. Going from polar to rectangular coordinates means plugging values into the formulas
x = r cos θ
y = r sin θ
Also, if the point in question lies on the x- or y-axis, there's no need for fancy formulas. We can translate visually.
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.