Unit 6, lesson 83

Polar Coordinates

How to locate a point by its distance from a fixed center and an angle, why the same point has many polar names, and how to switch between polar and rectangular coordinates.

  • Theory part 165
  • Practice part 166
  • Polar Curves

A lighthouse keeper does not say a ship is 3 kilometers east and 4 kilometers north. She says it is 5 kilometers away at a certain bearing. That is the idea behind polar coordinates. For anything that turns around a center, such as circles, spirals and orbits, distance and angle make the equations much simpler than xx and yy.

Key terms

Pole

Definition The pole is a fixed point OO of the plane chosen as the center of the polar coordinate system. When polar and rectangular coordinates are used together, the pole is the origin (0,0)(0, 0).

In plain words The pole is the point you measure everything from, like the lighthouse.

Example A point PP with polar coordinates (5,θ)(5, \theta) lies on the circle of radius 55 centered at the pole, whatever θ\theta is.

Polar axis

Definition The polar axis is a ray starting at the pole, from which angles are measured. In the standard setup it is the positive xx-axis.

In plain words It is the direction that counts as angle zero.

Example Every point with polar coordinates (r,0)(r, 0) and r>0r > 0 lies on the polar axis. The point (2,0)(2, 0) in polar coordinates is the point (2,0)(2, 0) in rectangular coordinates.

Polar coordinates and the angle in radians

Definition Let r>0r > 0 and let θ\theta be a real number. A point PP has polar coordinates (r,θ)(r, \theta) if its distance from the pole is rr and the ray from the pole through PP is obtained by turning the polar axis through the angle θ\theta. The angle is measured in radians, positive counterclockwise and negative clockwise. The pole itself has coordinates (0,θ)(0, \theta) for every real θ\theta.

In plain words First turn by θ\theta from the polar axis, then walk rr units out along that ray. Radians matter later, because the derivative formulas

ddθsin⁡θ=cos⁡θ,ddθcos⁡θ=−sin⁡θ\frac{d}{d\theta}\sin\theta = \cos\theta, \qquad \frac{d}{d\theta}\cos\theta = -\sin\theta

hold only when θ\theta is in radians.

Example The point (2,π3)(2, \frac{\pi}{3}) is 22 units from the pole on the ray at 6060 degrees above the positive xx-axis. The point (2,−π3)(2, -\frac{\pi}{3}) is its mirror image below the axis.

Negative radius

Definition For r<0r < 0, the point (r,θ)(r, \theta) is defined to be the point at distance ∣r∣\lvert r \rvert from the pole on the ray opposite to angle θ\theta, that is, on the ray of angle θ+π\theta + \pi. So

(−r,θ)and(r,θ+π)(-r, \theta) \quad \text{and} \quad (r, \theta + \pi)

name the same point for every rr and θ\theta.

In plain words A negative radius means face the direction θ\theta, then walk backwards.

Example The point (−2,π4)(-2, \frac{\pi}{4}) is 22 units from the pole in the direction 5π4\frac{5\pi}{4}. In rectangular coordinates it is (−2,−2)(-\sqrt{2}, -\sqrt{2}).

Nonunique coordinates

Definition For P≠OP \neq O, the pairs (r,θ)(r, \theta) that name PP are exactly

(r,θ+2nπ)and(−r,θ+(2n+1)π),n∈Z,(r, \theta + 2n\pi) \quad \text{and} \quad (-r, \theta + (2n + 1)\pi), \qquad n \in \mathbb{Z},

where (r,θ)(r, \theta) is any one of them. The pole is named by (0,θ)(0, \theta) for every real θ\theta.

In plain words Turning a full circle brings you back to the same ray, and a half turn with a walk backwards does too. So each point has infinitely many polar names.

Example The point (1,π2)(1, \frac{\pi}{2}), which is (0,1)(0, 1) in rectangular coordinates, can also be written (1,5π2)(1, \frac{5\pi}{2}), (1,−3π2)(1, -\frac{3\pi}{2}) or (−1,3π2)(-1, \frac{3\pi}{2}).

Converting between polar and rectangular

Definition If PP has polar coordinates (r,θ)(r, \theta) and rectangular coordinates (x,y)(x, y), then

x=rcos⁡θ,y=rsin⁡θ,x = r\cos\theta, \qquad y = r\sin\theta,

and consequently

r2=x2+y2,tan⁡θ=yxwhen x≠0.r^2 = x^2 + y^2, \qquad \tan\theta = \frac{y}{x} \quad \text{when } x \neq 0.

Going from a point (x,y)≠(0,0)(x, y) \neq (0, 0) to a name with r>0r > 0, take r=x2+y2r = \sqrt{x^2 + y^2}. If x≠0x \neq 0, the equation tan⁡θ=y/x\tan\theta = y/x has two solutions in [0,2π)[0, 2\pi), and you choose the one whose ray points into the quadrant of (x,y)(x, y). If x=0x = 0, then θ=π/2\theta = \pi/2 for y>0y > 0 and θ=3π/2\theta = 3\pi/2 for y<0y < 0.

In plain words Polar to rectangular is direct trigonometry. Rectangular to polar needs a picture, because the tangent alone cannot tell a point from its opposite.

Example The polar point (4,5π6)(4, \frac{5\pi}{6}) has rectangular coordinates

x=4cos⁡5π6=−23,y=4sin⁡5π6=2.x = 4\cos\frac{5\pi}{6} = -2\sqrt{3}, \qquad y = 4\sin\frac{5\pi}{6} = 2.

Going back the other way, for (−1,3)(-1, \sqrt{3}) we get r=1+3=2r = \sqrt{1 + 3} = 2 and tan⁡θ=−3\tan\theta = -\sqrt{3}. The candidates in [0,2π)[0, 2\pi) are 2π3\frac{2\pi}{3} and 5π3\frac{5\pi}{3}. The point is in the second quadrant, so θ=2π3\theta = \frac{2\pi}{3}.

Coordinate conventions

Definition Unless stated otherwise, the pole is the origin, the polar axis is the positive xx-axis, angles are in radians measured counterclockwise, and a polar curve r=f(θ)r = f(\theta) is the set of all points that have at least one pair of coordinates (r,θ)(r, \theta) satisfying the equation. When a single name is wanted for a point other than the pole, a common choice is r>0r > 0 and 0≤θ<2π0 \le \theta < 2\pi (some books use −π<θ≤π-\pi < \theta \le \pi).

In plain words A point is on a polar curve if some name of it fits the equation, even if other names do not.

Example Take the curve r=cos⁡2θr = \cos 2\theta and the point (0,1)(0, 1), which has the polar name (1,π2)(1, \frac{\pi}{2}). That name does not fit, since cos⁡π=−1≠1\cos \pi = -1 \neq 1. The same point is also (−1,3π2)(-1, \frac{3\pi}{2}), and cos⁡3π=−1\cos 3\pi = -1 matches r=−1r = -1. So (0,1)(0, 1) is on the curve. Testing only one name would have given the wrong answer.

Check yourself

  1. Convert the polar point (3,3π2)(3, \frac{3\pi}{2}) to rectangular coordinates.
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The conversion formulas give

x=3cos⁡3π2=0,y=3sin⁡3π2=−3,x = 3\cos\frac{3\pi}{2} = 0, \qquad y = 3\sin\frac{3\pi}{2} = -3,

so the point is (0,−3)(0, -3).

  1. Give polar coordinates of (−2,−2)(-2, -2) with r>0r > 0 and 0≤θ<2π0 \le \theta < 2\pi, and one more name with r<0r < 0.
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The distance is r=8=22r = \sqrt{8} = 2\sqrt{2} and the point is in the third quadrant, so θ=5π4\theta = \frac{5\pi}{4}. Another name is (−22,π4)(-2\sqrt{2}, \frac{\pi}{4}).

  1. Write the polar curve r=2sin⁡θr = 2\sin\theta in rectangular form.
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Multiply by rr to get r2=2rsin⁡θr^2 = 2r\sin\theta, so x2+y2=2yx^2 + y^2 = 2y, which is x2+(y−1)2=1x^2 + (y - 1)^2 = 1. Multiplying by rr adds no new points, because the pole is already on the curve at θ=0\theta = 0.

How this lesson connects

It also uses Coordinates and radians and trigonometry.