Polar Coordinates

Try this

Pick a point $O$ in the plane (the pole), draw a ray $O x$ from it (the polar axis), and fix a unit of length and a positive direction for angles (counterclockwise). Now the position of any point $P$ is described by two numbers: $∣ O P ∣ = ρ$ (the radius) and the angle $θ$ you turn through from $O x$ to $O P$ (the polar angle). The pair $(ρ, θ)$ is called the polar coordinates of $P$ .

Drag $P$ slowly once around $O$ . Which number changes, and which stays the same? When you are back where you started, is $θ$ equal to $0$ or $2 π$ ?
Use the slider to set $θ$ to $\frac{7 π}{3}$ ( $42 0^{\circ}$ ). Where is $P$ ? Compare it with $θ = \frac{π}{3}$ .
Keep $θ$ fixed and pull $ρ$ from $3$ to $- 3$ . Along which line does $P$ move? Where does it end up?
Turn on “Other names of this point”. What do the circled positions have in common?
Set $ρ$ to $0$ and change $θ$ however you like. Does $P$ move?

Put the pole at the origin and the polar axis along the positive $x$ -axis, and the two coordinate systems line up. Whether $ρ$ is positive or negative, we always have

x = ρ cos θ, y = ρ sin θ,

and conversely $ρ^{2} = x^{2} + y^{2}$ , $tan θ = \frac{y}{x} (x \neq = 0)$ . The first pair of formulas is always single-valued: given polar coordinates, there is exactly one rectangular point. The second pair makes you choose: whether $ρ$ is positive or negative, and which $θ$ you take, has to be stated.

Predict first

In which quadrant is the point with polar coordinates $(- 2, \frac{π}{3})$ ?

FirstSecondThirdFourth

Choose an answer first; then you can see the result.

Why

“One point, many names” is not a flaw: it is a freedom you can use. Here is a concrete conversion.

Example: convert the rectangular point (−2√3, 2) to polar coordinates

Step 1 / 5

First the radius
$ρ^{2} = x^{2} + y^{2} = 12 + 4 = 16$ , so take $ρ = 4$ . $P$ must be on the circle of radius $4$ ; where on the circle depends on the angle.

$ρ = (- 23)^{2} + 2^{2} = 4$
Copy the arctangent blindly
Check the quadrant, then pick the angle
The “wrong” angle works too
One point, many names

This freedom pays off at the next stop: often only one of a point’s names satisfies a curve’s polar equation and the others do not. To decide whether a point lies on a curve, you have to think of all its names.

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: Every point other than the pole has exactly one pair of polar coordinates

With this rule, dragging $P$ always gives a unique pair $(ρ, θ)$ : after a full turn, $θ$ comes back to $0$ instead of becoming $2 π$ .

Practice

Exercise 1Basic

Point $A$ has polar coordinates $(6, \frac{4 π}{3})$ . What is its rectangular $x$ -coordinate?

Exercise 2Basic

Convert the rectangular point $(1, - 1)$ to polar coordinates with $ρ > 0$ and $0 \leq θ < 2 π$ . If $θ = k π$ , what is $k$ ?

Exercise 3Advanced

In a polar coordinate system, $P_{1} (2, \frac{π}{6})$ and $P_{2} (4, \frac{π}{2})$ . Find $∣ P_{1} P_{2} ∣^{2}$ .

Exercise 4Challenge

$O$ is the pole, $A (3, \frac{π}{12})$ , $B (- 4, \frac{17 π}{12})$ . The area of $△ O A B$ is $k 3$ . Find $k$ .

Try this

Why

Example: convert the rectangular point (−2√3, 2) to polar coordinates

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Using $\theta=\arctan\dfrac yx$ without looking at the quadrant

Seeing $\theta$ in the first quadrant and assuming the point is there too