Trigonometry · step 12 of 13

Polar coordinates & converting to Cartesian

Polar coordinates locate a point by distance r from the origin and angle θ from the positive x-axis. Conversion is x = r cos θ, y = r sin θ one way, and r = √(x²+y²), θ = arctan(y/x) the other — with the arctan needing a quadrant adjustment, since it cannot distinguish (1,1) from (−1,−1).

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Polar to Cartesian is directx = r cos θ and y = r sin θ. No ambiguity in this direction.
Cartesian to polar needs care with θr = √(x²+y²) is unambiguous, but arctan(y/x) returns a value in (−π/2, π/2) — add π when x < 0.
Note that polar representation is not unique(r, θ) = (r, θ + 2π), and (−r, θ) = (r, θ + π). Questions usually want r ≥ 0 and θ in a stated range.
Convert equations by substitutionA circle x² + y² = 9 becomes simply r = 3; a line through the origin becomes θ = constant.

Worked example

Convert the Cartesian point (−1, √3) to polar coordinates with r ≥ 0 and 0 ≤ θ < 2π.

r = √(1 + 3) = 2.
arctan(√3/(−1)) = arctan(−√3) = −π/3, but the point is in the second quadrant.
Add π to correct the quadrant: θ = −π/3 + π = 2π/3.
Check: 2cos(2π/3) = 2(−1/2) = −1 ✓ and 2sin(2π/3) = 2(√3/2) = √3 ✓.

Answer. (r, θ) = (2, 2π/3).

Where marks get dropped

These are the specific errors that cost credit on polar coordinates & converting to cartesian questions — QED's rubric penalises each of them separately.

Trusting arctan for the quadrant. It cannot distinguish the second quadrant from the fourth — always check the signs of x and y.
Assuming polar coordinates are unique. Adding 2π to θ, or negating r and adding π, names the same point.
Forgetting that r = 0 makes θ arbitrary — the origin has no well-defined angle.

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Polar coordinates & converting to Cartesian — frequently asked questions

How do I get θ reliably?

Use the two-argument arctangent, atan2(y, x), which uses the signs of both coordinates to place the angle in the correct quadrant. By hand, compute the reference angle and adjust.

What curve is r = 2cos θ?

A circle of radius 1 centred at (1, 0). Multiplying by r gives r² = 2r cos θ, i.e. x² + y² = 2x, which completes the square to (x−1)² + y² = 1.

Why use polar coordinates?

Because circular and rotational symmetry becomes trivial: a circle is r = a and a spiral is r = aθ, both far simpler than their Cartesian forms.

The rest of Trigonometry

Triangles, the unit circle, identities and periodic functions. Each subtopic below has its own method, worked example and mark-losing traps.

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