How this instrument works
A point in the plane can be pinned down two ways: by how far right and up it sits, its Cartesian coordinates (x, y), or by how far out and which way it points from the origin, its polar coordinates (r, θ). This page turns the second description into the first. Start from the unit circle, where cosine and sine are defined in the first place: the point at angle θ on a circle of radius 1 sits at (cos θ, sin θ) by definition, cosine giving the horizontal coordinate and sine the vertical one. Scale that circle up to radius r and both coordinates scale with it, which is exactly the pair the calculator evaluates: x = r·cos(θ), y = r·sin(θ).
This direction of the conversion is the easy one. Fixing r and θ pins down x and y with a single pass through cosine and sine — no sign-checking and no choosing between candidate quadrants, because the direction is handed to the formula complete rather than reconstructed afterward. The reverse trip, recovering an angle from an already-computed x and y, needs an extra rule to put the point back in the right quadrant, precisely because a plain coordinate pair no longer carries direction as an explicit label the way θ does here.
Two limits anchor the picture. Sweep θ through a full 360° while holding r fixed and the point traces the entire circle of that radius exactly once, since the two formulas are nothing more than a parametrisation of a circle; at θ = 0° it lands on (r, 0), and at θ = 90° it lands on (0, r), because one of cosine or sine vanishes exactly there. At r = 0, every angle collapses onto the same point, the origin — a radius of nothing leaves no room for direction to matter.
- Enter the distance from the origin into r (radius) — zero or positive, in whatever unit you want x and y reported in.
- Enter the direction into θ (angle); the unit control beside the field switches between degrees, radians, and turns, and defaults to degrees.
- Read x for the point's horizontal position and y for its vertical position, both measured from the origin.
- Set θ to 0° or 90° as a sanity check — the point should land exactly on the x-axis or y-axis before you trust an in-between angle.
Worked example — plotting the point at r = 5, θ ≈ 53.13°
Suppose a hand sketch or a polar equation specifies a point 5 units from the origin at an angle of 53.13010235415598°, the field's default setting, which the sheet stores internally as 0.9272952180016122 radians before evaluating the trig functions. It returns x = 5 × cos(0.9272952180016122) = 3.0000000000000004 and y = 5 × sin(0.9272952180016122) = 3.9999999999999996 — the trailing digits are ordinary floating-point rounding, and the point to plot is (3, 4).
That answer checks out without a calculator at all: 53.13° is the angle whose cosine is 3⁄5 and whose sine is 4⁄5, so 5 × 3⁄5 = 3 and 5 × 4⁄5 = 4 exactly, and 3² + 4² = 9 + 16 = 25 = 5², confirming the point really does sit 5 units out. Mark it on graph paper, three squares right and four up, and it lands exactly where a ruler and protractor would put it.
Questions
What do r and θ represent in this conversion?
r is the point's straight-line distance from the origin, always zero or positive, and θ is its direction, measured counterclockwise from the positive x-axis. Together (r, θ) is the polar description of the same point that (x, y) describes in rectangular coordinates; this page turns the first pair into the second.
Why does x use cosine and y use sine, not the reverse?
It follows straight from how sine and cosine are defined: on the unit circle, the point at angle θ sits at (cos θ, sin θ), cosine fixing the horizontal coordinate and sine the vertical one. Scaling that circle to radius r scales both coordinates by r, giving exactly x = r cos θ and y = r sin θ.
Do I need to convert θ to radians before entering it?
No — the unit control next to the angle field accepts degrees, radians, or turns, and the sheet converts to radians internally before calling cosine and sine, since those functions are defined on radian measure. Enter whatever unit your source data already uses.
What happens if r is 0, or if θ goes past 360°?
An r of 0 always returns the origin, (0, 0), whatever θ is set to, since a radius of nothing leaves no room for direction to matter. Angles beyond 360° or below 0° work fine too — cosine and sine are periodic, so θ and θ + 360° land on the identical point.
How is this different from converting a complex number's polar form to a + bi?
The arithmetic is identical, but the second coordinate means something different. Here y is an ordinary vertical distance on the plane, while in a + bi form, b is the coefficient multiplying i and follows complex-number arithmetic once it's used further, not plain vector addition.
Why doesn't this direction need a rule like atan2, the way the reverse conversion does?
Because θ arrives as a direct input here, there is no ratio to rebuild it from and so no quadrant to guess. That extra machinery only becomes necessary going the other way, from (x, y) back to (r, θ), where the angle has to be recovered from the coordinates rather than simply read off the dial.