Key takeaway: Polar coordinates name a location with a radius r and a turn θ, the pair (r, θ), instead of a horizontal step and a vertical step.
Grade 12 – PrecalculusN-CN.B.4Geometry and Measurement
Definition of Polar Coordinates
Polar coordinates describe position the way a radar screen does: how far away, and in which direction. The fixed center is the pole, and the starting ray is the polar axis, which lies along the positive x-axis.
A pair \((r,\theta)\) says: turn through \(\theta\), then walk out \(r\) units. Unlike rectangular coordinates, one location owns many polar names. Adding a full turn to the angle gives the same point, and a negative \(r\) means walk backward from the heading. Circles and spirals have simple polar equations that are messy in x and y.
Formula and notation
Converting between the two systems:
\[x=r\cos\theta,\quad y=r\sin\theta,\qquad r=\sqrt{x^2+y^2},\quad \tan\theta=\frac{y}{x}.\]
For the reverse direction, the quadrant of \((x,y)\) decides which of the two possible angles is correct.
Key facts
- The pole itself is (0, θ) for any angle θ.
- In polar coordinates, (r, θ) and (r, θ + 360°) name the same point.
- The pair (−r, θ) is the same point as (r, θ + 180°).
- The equation r = 5 is a circle; θ = 40° is a straight line through the pole.
Polar Coordinates example
Convert the polar point \((7,210^{\circ})\) to rectangular coordinates.
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Horizontal: \(x=7\cos 210^{\circ}=7(-\frac{\sqrt{3}}{2})\approx -6.06\).
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Vertical: \(y=7\sin 210^{\circ}=7(-\frac{1}{2})=-3.5\).
The rectangular point is \((-\frac{7\sqrt{3}}{2},-\frac{7}{2})\approx(-6.06,-3.5)\).
Common mistake
Watch out
Going back, Ines takes \((-6.06,-3.5)\), computes \(\tan^{-1}(\frac{-3.5}{-6.06})=30^{\circ}\) and names the point \((7,30^{\circ})\).
Why it is wrong
Negative x and negative y mean quadrant III, not quadrant I. The calculator’s 30° must become \(30^{\circ}+180^{\circ}=210^{\circ}\). Sketch the quadrant before naming the angle in polar coordinates.
Related terms and lessons
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