Polar Coordinates
In short: Polar coordinates pin down a location with a range r from the center and a turn θ counterclockwise off the positive x-axis, recorded as (r, θ) rather than (x, y).
PrecalculusGeometryN-CN.B.4
What is Polar Coordinates?
Polar coordinates give directions the way a radar does: how far, and in what direction. A point is named by the pair \((r, \theta)\), where \(r\) is the distance from the origin and \(\theta\) is the angle of rotation.
The links to rectangular coordinates are \(x = r\cos\theta\), \(y = r\sin\theta\), \(r^2 = x^2 + y^2\) and \(\tan\theta = \frac{y}{x}\). Unlike \((x, y)\), one location owns many names: \((r, \theta)\) and \((r, \theta + 2\pi)\) agree, and so do \((r, \theta)\) and \((-r, \theta + \pi)\).
How to spot it
- A situation involves circles, spirals or rotation about a center, such as a radar sweep or a sprinkler.
- A point is given as (r, θ) with an angle in degrees or radians.
- A conversion between (x, y) and (r, θ) is requested.
- An equation such as r = 4 or θ = π/6 draws a circle or a straight line through the origin.
The method
- To go from polar to rectangular, compute \(x = r\cos\theta\) and \(y = r\sin\theta\).
- To go the other way, compute \(r = \sqrt{x^2 + y^2}\).
- Find \(\theta\) from the reference angle \(\tan^{-1}\left|\frac{y}{x}\right|\), then place it in the correct quadrant.
- Check by converting your answer back.
Polar Coordinates: solved example
Problem
Convert \((10, \frac{7\pi}{6})\) to rectangular form, and convert \((-2, 2)\) to polar form.
Solution
First point: \(x = 10\cos 210^{\circ} = -5\sqrt{3} \approx -8.66\) and \(y = 10\sin 210^{\circ} = -5\).
Second point: \(r = \sqrt{4 + 4} = 2\sqrt{2}\), and since it is in Quadrant II with reference angle \(45^{\circ}\), \(\theta = \frac{3\pi}{4}\). The answers are \((-5\sqrt{3}, -5)\) and \((2\sqrt{2}, \frac{3\pi}{4})\).
Try it
Question 1
Convert \((3, \frac{\pi}{2})\) to rectangular coordinates.
Show the answer
\(x = 3\cos\frac{\pi}{2} = 0\) and \(y = 3\sin\frac{\pi}{2} = 3\), giving \((0, 3)\).
Question 2
Convert \((-6, 0)\) to polar coordinates with \(r > 0\).
Show the answer
\(r = 6\), and the point lies on the negative x-axis, so \(\theta = \pi\): \((6, \pi)\).
Polar Coordinates vs. Polar Form of a Complex Number
Both use a distance and an angle, but their purpose differs. The coordinates here simply name a location on a plane and describe curves such as circles and spirals. The complex version attaches that pair to a number that can be added, multiplied and raised to powers.
Keep going
Lessons
Worksheets and quizzes
- Polar Form and De Moivre’s Theorem – Worksheet
- Polar Form and De Moivre’s Theorem – Quiz
Related words
- Polar Form of a Complex Number
- Unit Circle
- Coordinate Plane
- Ordered Pair
- Radian
- Parametric Equations
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