Theorem

The parametric equations are

The same shape can be defined in polar coordinates by the equation

Proof

Start from r = 1 + cosθ. From Polar_coordinates#Cartesian_and_cylindrical and Trigonometric_identity#Double-angle_formulas we see that we get the cartesian coordinates: This article describes some of the common coordinate systems that appear in elementary mathematics. ... In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...

Another proof

Equations (1) and (2) define a cardioid whose cuspidal point is (−1/2, 0). To convert to polar, the cusp should preferably be at the origin, so add 1/2 to the abscissa: The origin of something (from the Latin origo, beginning) is where it came from, in the sense of a physical location or a metaphysical source. ... Abscissa means the x coordinate on an (x, y) graph; the input of a mathematical function against which the output is plotted. ...

The polar radius ρ(θ) is given by RADIUS (Remote Authentication Dial In User Service) is an AAA (authentication, authorization and accounting) protocol for applications such as network access or IP mobility. ...

Expand:

Simplify by noticing that

Thus,

Then, since

it follows that

quod erat demonstrandum. For other meanings of the abbreviation QED, see QED. Q. E. D. is an abbreviation of the Latin phrase quod erat demonstrandum (literally, that which was to be demonstrated). This is a translation of the Greek oper edei deixai which was used by many early mathematicians including Euclid and Archimedes. ...

Area derivation

The objective is to integrate the area of the cardioid

r = 1 − cosθ.

The integral is In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ...

.

Integration with respect to dr yields

Distribute the integral among the three terms, and integrate the first two, to obtain

The second term vanishes, and integrating the third term yields

The last term within brackets vanishes, so that

Cardioids of any size are all similar to each other, so increasing the cardioid's linear size by a factor of a increases the cardioid's areal size by a factor of a²,    Q.E.D.   (return to article) In geometry, the cardioid is an epicycloid which has one and only one cusp. ...