Represent a point with radius and angle

Try

Locate a point 4 units from the origin along a ray at angle π/3. Its Cartesian coordinates are x=4cos(π/3)=2 and y=4sin(π/3)=2√3. The polar pair (4,π/3) describes the same point using radius and angle rather than horizontal and vertical components.

Predict

Predict the location of (−4,π/3). A negative radius reverses direction along the ray; it does not place a point at a negative physical distance. Its distance from the origin is 4, and its Cartesian coordinates are (−2,−2√3). A positive-radius representation is (4,4π/3).

Experiment

Plot these pairs and group those describing the same point.

Polar pair	Cartesian point
(3,0)	(3,0)
(3,2π)	(3,0)
(−3,π)	(3,0)
(0,π/4)	(0,0)
(0,π)	(0,0)

The first three show two kinds of nonuniqueness: full-turn changes and radius reversal with a half-turn. The last two show that every angle represents the origin when radius is zero.

Understand

The conversion formulas are x=r cos θ and y=r sin θ. Therefore r²=x²+y², and the nonnegative radius is √(x²+y²). If negative radii are allowed, either sign can be paired with a suitable angle. The actual distance from the origin is |r|.

To recover an angle for a nonzero point, inspect the signs of x and y. For (−2,2), the ratio y/x is −1 and ordinary arctangent returns −π/4. That is the wrong quadrant for a positive radius. Choose 3π/4 instead; then r=2√2 gives both original components. The tangent ratio alone does not distinguish opposite quadrants.

For x=0, dividing y by x is invalid, but the point can still be represented: a positive vertical point has θ=π/2 and a negative one θ=3π/2 with positive radius. At the origin, angle is not uniquely determined. A conversion algorithm must treat these cases rather than silently dividing by zero.

You may choose a canonical angle interval such as [0,2π) to report one representation. This is a convention, not a claim that other coterminal representations are wrong. When an answer requests a particular interval or a positive radius, follow that restriction explicitly.

Apply

Use the practice to verify conversions in both directions. For a written negative-radius representation, substitute into both coordinate formulas. A correct radius without the correct quadrant is incomplete. Keep exact radicals when the point’s distance is irrational; the choice and written tasks do not ask you to approximate symbolic values.

Continue

When radius varies with angle, the pairs trace a polar curve. Its graph can repeat points, pass through the origin or move downward while radius increases, so tracing requires both coordinates together.