Polar Coordinates

Polar coordinates name a point by its distance from the origin and its angle, written (r, \theta), instead of by x and y. To convert from polar to rectangular, use x = r \cos \theta and y = r \sin \theta. To go back, use r = \sqrt{x^2 + y^2} and \tan \theta = \frac{y}{x}, checking the quadrant.

Before this lesson: You should know The Unit Circle, Radians and Coterminal Angles.

What Polar Coordinates Mean

Rectangular coordinates tell you how far to go right and up. Polar coordinates tell you which way to face and how far to walk. The origin is called the pole, and the positive x-axis is the polar axis.

In (r, \theta), \theta is the angle measured counterclockwise from the polar axis. The number r is the distance from the pole along that direction. A polar grid uses circles for r and rays for \theta.

One point, many names

In rectangular coordinates, each point has one name. In polar coordinates, each point has many. Adding 2\pi to the angle lands on the same ray, so (4, \frac{\pi}{3}) is also (4, \frac{7\pi}{3}).

A negative r means walk backward, in the opposite direction. So (-4, \frac{4\pi}{3}) is the same point too, because \frac{4\pi}{3} points the opposite way from \frac{\pi}{3}.

Conversion Formulas

Polar to rectangular:

x = r \cos \theta, \quad y = r \sin \theta

Rectangular to polar:

r = \sqrt{x^2 + y^2}, \quad \tan \theta = \frac{y}{x}

These come from a right triangle with legs x and y and hypotenuse r. When you use \tan \theta = \frac{y}{x}, the calculator only gives angles in Quadrants I and IV. Always check which quadrant (x, y) is in, and adjust.

Worked Examples

Example 1: Polar to rectangular

Convert (4, \frac{\pi}{3}) to rectangular coordinates.

x = 4 \cos \frac{\pi}{3}

x = 4 \cdot \frac{1}{2} = 2

y = 4 \sin \frac{\pi}{3}

y = 4 \cdot \frac{\sqrt{3}}{2} = 2\sqrt{3}

Answer: (2, 2\sqrt{3}), which is about (2, 3.46). This matches the picture above.

Example 2: Rectangular to polar

Convert (-3, 3) to polar coordinates with r \gt 0 and 0 \le \theta \lt 2\pi.

r = \sqrt{(-3)^2 + 3^2}

r = \sqrt{9 + 9} = \sqrt{18}

r = \sqrt{9 \cdot 2} = 3\sqrt{2}

\tan \theta = \frac{3}{-3} = -1

The reference angle is \frac{\pi}{4}, and (-3, 3) is in Quadrant II (see reference angles)

\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}

Answer: (3\sqrt{2}, \frac{3\pi}{4}). A calculator gives \tan^{-1}(-1) = -\frac{\pi}{4}, which points into Quadrant IV, so it is the wrong direction.

Example 3: Other names for the same point

Give two other polar names for (2, \frac{\pi}{6}): one with a positive r, and one with a negative r.

Add 2\pi to the angle, using a common denominator: \frac{\pi}{6} + \frac{12\pi}{6} = \frac{13\pi}{6}, so (2, \frac{13\pi}{6})

\frac{\pi}{6} + \frac{6\pi}{6} = \frac{7\pi}{6}

(-2, \frac{7\pi}{6})

Answer: (2, \frac{13\pi}{6}) and (-2, \frac{7\pi}{6}). Other answers, like (2, -\frac{11\pi}{6}), are also correct.

Example 4: Convert a polar equation

Write r = 6 \sin \theta in rectangular form, and describe its graph.

r^2 = 6r \sin \theta

x^2 + y^2 = 6y

x^2 + y^2 - 6y = 0

Complete the square for y (see completing the square): add 9 to both sides to get x^2 + y^2 - 6y + 9 = 9

x^2 + (y - 3)^2 = 9

Answer: x^2 + (y - 3)^2 = 9, a circle with center (0, 3) and radius 3 (see equation of a circle).

Example 5: Word problem with a radar screen

A radar screen shows a plane 50 km from the tower, at an angle of 120^\circ counterclockwise from due east. How far east or west and how far north or south of the tower is the plane?

(50, 120^\circ)

x = 50 \cos 120^\circ

x = 50 \cdot (-\frac{1}{2}) = -25

y = 50 \sin 120^\circ

y = 50 \cdot \frac{\sqrt{3}}{2} = 25\sqrt{3} \approx 43.30

Answer: The plane is 25 km west and about 43.30 km north of the tower. A negative x means west.

Common Mistakes

Related Methods

Degrees or radians

Polar angles can be in degrees or radians. Use whichever the problem uses, and set your calculator to match.

Converting equations both ways

To turn a rectangular equation into polar form, replace x with r \cos \theta and y with r \sin \theta. For example, x^2 + y^2 = 25 becomes r^2 = 25, so r = 5.

Complex numbers

A complex number a + bi can be plotted at (a, b). Its polar form uses the same r and \theta, which leads to the trigonometric form of complex numbers.

Practice Problems

Give exact answers when possible. Use r \gt 0 and 0 \le \theta \lt 2\pi unless the problem says otherwise.

1. Convert (6, \frac{\pi}{6}) to rectangular coordinates.
2. Convert (5, \frac{3\pi}{2}) to rectangular coordinates.
3. Convert (-4, \frac{\pi}{4}) to rectangular coordinates.
4. Convert (0, -7) to polar coordinates.
5. Convert (2, -2\sqrt{3}) to polar coordinates.
6. Convert (-5, -12) to polar coordinates. Round \theta to the nearest hundredth of a radian.
7. Give two other polar names for (3, \frac{2\pi}{3}): one with a negative angle and one with a negative r.
8. Write x^2 + y^2 = 25 in polar form.
9. Write r = 4 \cos \theta in rectangular form.
10. A lighthouse spots a boat 8 km away at 210^\circ counterclockwise from due east. How far west and south of the lighthouse is the boat?

x = 6 \cos \frac{\pi}{6} = 6 \cdot \frac{\sqrt{3}}{2} = 3\sqrt{3}

y = 6 \sin \frac{\pi}{6} = 6 \cdot \frac{1}{2} = 3

x = 5 \cos \frac{3\pi}{2} = 5 \cdot 0 = 0

y = 5 \sin \frac{3\pi}{2} = 5 \cdot (-1) = -5

x = -4 \cos \frac{\pi}{4} = -4 \cdot \frac{\sqrt{2}}{2} = -2\sqrt{2}

y = -4 \sin \frac{\pi}{4} = -4 \cdot \frac{\sqrt{2}}{2} = -2\sqrt{2}

r = \sqrt{0^2 + (-7)^2} = \sqrt{49} = 7

r = \sqrt{2^2 + (-2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4

\theta = 2\pi - \frac{\pi}{3} = \frac{5\pi}{3}

r = \sqrt{(-5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13

\tan^{-1}(\frac{12}{5}) \approx 1.18

\theta \approx \pi + 1.176 \approx 4.32

r^2 = 25

r = 5

r^2 = 4r \cos \theta

x^2 + y^2 = 4x

x^2 - 4x + y^2 = 0

x^2 - 4x + 4 + y^2 = 4

(x - 2)^2 + y^2 = 4

x = 8 \cos 210^\circ = 8 \cdot (-\frac{\sqrt{3}}{2}) = -4\sqrt{3} \approx -6.93

y = 8 \sin 210^\circ = 8 \cdot (-\frac{1}{2}) = -4

FAQ

How do you convert polar to rectangular coordinates?

Use x = r \cos \theta and y = r \sin \theta. Multiply the distance by the cosine and sine of the angle.

How do you convert rectangular to polar coordinates?

Find r = \sqrt{x^2 + y^2}. Then find the angle from \tan \theta = \frac{y}{x}, and adjust it for the quadrant the point is in.

Can r be negative in polar coordinates?

Yes. A negative r means you go the opposite way from the angle. The point (-r, \theta) is the same as (r, \theta + \pi).

Why do points have more than one polar name?

Angles repeat every 2\pi, and a negative r can reach the same point from the opposite angle. So every point has endless polar names.

Summary

• A polar point (r, \theta) is a distance r from the pole at angle \theta.
• Polar to rectangular: x = r \cos \theta, y = r \sin \theta.
• Rectangular to polar: r = \sqrt{x^2 + y^2}, \tan \theta = \frac{y}{x}, then check the quadrant.
• Each point has many polar names: add 2\pi, or use -r with \theta + \pi.
• Convert equations with x^2 + y^2 = r^2, x = r \cos \theta and y = r \sin \theta.

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