Polar to rectangular converter

This capability takes a finite polar radius r and angle θ—in degrees by default or in radians when you ask—and returns the rectangular pair x equals r times cosine of θ and y equals r times sine of θ as deterministic JSON. Negative radii are rejected by default so classroom distance workflows stay safe; enable allow_negative_r when you need the signed-radius convention used in some analytic geometry texts. There is no computer-algebra system, no plotter, no random seed, and no wall-clock dependency: the free browser tool and the prepaid API path share one pure solve function, so a homework check and a production script agree after fixed decimal rounding.

What polar and rectangular coordinates mean

The formulas are deterministic after at most one unit conversion—no series is re-implemented in userland, no sampling is required, and the result does not depend on locale, time zone, or network state—so graders, shaders, and API clients see the same rounded numbers every time.

How to call the API and what comes back

Use it from homework pipelines, interview-prep sandboxes, robotics and game tooling, survey and navigation scripts, or any workflow that needs reproducible polar-to-cartesian conversion without installing a computer-algebra package on every machine. Send a JSON body with r and theta. Field r is the polar radius; aliases radius, rho, and the ρ character map to the same meaning. Field theta is the polar angle; aliases angle, the θ character, and t also work. Optional angle_unit is degrees by default or radians when you pass radians (aliases unit and angle_mode). Optional allow_negative_r defaults to false; set it true only when your pipeline intentionally uses signed radii. On success the response echoes r, theta, angle_unit, and allow_negative_r; adds theta_radians and theta_degrees for convenience; returns x and y as rounded numbers; and includes method set to polar_to_rectangular plus a short formula block documenting the definitions. Failed validation—missing r or theta, a non-finite value, an unknown unit, a magnitude past published limits, or a negative r while negative radii are disallowed—returns invalid_input and is never charged. Pricing is a flat $0.002 per successful request with unit item and zero marginal unit price.

Practical tips, negative radii, and related searches

Prefer well-known classroom pairs when checking by hand: r equals one at zero, thirty, forty-five, sixty, and ninety degrees match unit-circle tables after rounding. When working in radians, common inputs are zero, π over six, π over four, π over three, and π over two; set angle_unit to radians so the conversion step is skipped. Keep r non-negative unless you explicitly understand the signed-radius convention: a negative r with allow_negative_r true uses the same multiplications and lands at the opposite ray, which is equivalent to absolute r at θ plus π. If your data source sometimes emits negative distances by mistake, leave the default guard on and treat invalid_input as a data-quality signal rather than silently flipping the point. Absolute r and theta magnitudes are capped so intermediate IEEE doubles stay finite on the Worker and in the browser. This landing targets people searching for a polar to rectangular converter or an r theta to x y cartesian tool in degrees or radians; it is not a rectangular-to-polar inverter and does not solve polar equations. Failed tasks that return invalid_input do not consume prepaid balance, which keeps exploratory scripts cheap when learners probe bad radii on purpose. After a successful call, a useful sanity check is that the Euclidean length of (x, y) equals the absolute value of r within floating tolerance, and that the angle of the vector matches θ modulo two π when r is positive.