Polar to Cartesian converter

Type r and then θ. It reads them the way they're written in books: `2, 60`, `r = 2, θ = 60°`, `2∠60°` and `2 cis 60°` are all the same point. The x and y values appear underneath, and the working shows each cosine and sine.

Angles are read as degrees unless the Angles setting says radians. Anything written with π, like `π/3` or `3pi/4`, is read as radians whatever the setting, and so is anything followed by rad.

Drop a line from the point to the x axis and you have a right-angled triangle with r as its hypotenuse. The side along the x axis is r cos θ and the side going up is r sin θ. So x = r cos θ and y = r sin θ.

For r = 2 at 60°, cos 60° is 0.5 and sin 60° is about 0.8660, so the point is (1, 1.7321). The signs take care of themselves. At 150° the cosine is negative, and x comes out negative without any adjustment.

A negative r is allowed. It means go the opposite way, so (−2, 0°) is the same point as (2, 180°), and the working says so.

Angles wrap round. 420° is 60° plus a full turn and lands on the same point, which is why one point has endless polar forms but only one Cartesian one.

At right angles the converter gives exact answers. cos 90° is printed as 0, not as the 6.1 × 10⁻¹⁷ a calculator's rounding can leave behind.

Grégoire de Saint-Vincent and Bonaventura Cavalieri each worked with the idea in the first half of the 17th century. Isaac Newton described the conversion in his Method of Fluxions, written in 1671 and not published until 1736, and Jacob Bernoulli used a pole and a polar axis in Acta Eruditorum in 1691. The name came later, from 18th-century Italian writers, and reached English in George Peacock's 1816 translation of Lacroix's calculus textbook.