Cardioid proofs

From Wikipedia, the free encyclopedia

This mathematics article is devoted entirely to providing mathematical proofs and support for claims and statements made in the article Cardioid. This article is currently an experimental vehicle to see how we might be able to provide proofs and details for math articles without cluttering up the main article itself. See Wikipedia:WikiProject Mathematics/Proofs for current discussion. This article is "experimental" in that it is a proposal for one way that we might be able to deal with expressing proofs.

1 Theorem
- 1.1 Proof
- 1.2 Another proof
2 Area derivation

[edit] Theorem

The parametric equations are

$x(\theta) = \cos \theta + {1 \over 2} +{1 \over 2} \cos 2 \theta, \qquad \qquad (1)$

$y(\theta) = \sin \theta + {1 \over 2} \sin 2 \theta. \qquad \qquad (2)$

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

$\rho(\theta) = 1 + \cos \theta. \$

[edit] 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:

$x = r \cos \theta = \cos \theta + \cos^2 \theta = \cos \theta + (1 + \cos 2 \theta)/2\;$

$y = r \sin \theta = \sin \theta + \cos \theta \sin \theta= \sin \theta + (\sin 2 \theta)/2\,$

[edit] 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:

$x(\theta) = {1 \over 2} + \cos \theta + {1 \over 2} \cos 2 \theta,$

$y(\theta) = \sin \theta + {1 \over 2} \sin 2 \theta.$

The polar radius $ρ(θ)$ is given by

$\rho(\theta) = \sqrt{x^2(\theta) + y^2(\theta)}$

$= \sqrt{\left( {1 \over 2} + \cos \theta + {1 \over 2} \cos 2 \theta \right)^2 + \left( \sin \theta + {1 \over 2} \sin 2 \theta \right)^2 }.$

Expand:

$\rho = \sqrt{ {1 \over 4} + \cos^2 \theta + {1 \over 4} \cos^2 2 \theta + \cos \theta + {1 \over 2} \cos 2 \theta + \cos \theta \cos 2 \theta + \sin^2 \theta + {1 \over 4} \sin^2 2 \theta + \sin \theta \sin 2 \theta}.$

Simplify by noticing that

$\cos^2 \theta + \sin^2 \theta = 1, \qquad \qquad \mbox{(trigonometric identity)}$

${1 \over 4} \cos^2 2 \theta + {1 \over 4} \sin^2 2 \theta = {1 \over 4}, \qquad \qquad \mbox{(variation of the above)}$

$\cos \theta \cos 2 \theta + \sin \theta \sin 2 \theta = \cos (\theta - 2 \theta) = \cos -\theta = \cos \theta. \$

Thus,

$\rho = \sqrt{ {1 \over 4} + 1 + {1 \over 4} + 2 \cos \theta + {1 \over 2} \cos 2 \theta }$

$= \sqrt{ {3 \over 2} + {4 \over 2} \cos \theta + {1 \over 2} \cos 2 \theta }$

$= \sqrt{ {3 + 4 \cos \theta + \cos 2 \theta \over 2}}.$

Then, since

$\cos 2 \theta = \cos^2 \theta - \sin^2 \theta = 2 \cos^2 \theta - 1, \qquad \qquad \mbox{(trigonometric identity)}$

it follows that

$\rho = \sqrt{ {3 + 4 \cos \theta + 2 \cos^2 \theta - 1 \over 2}} = \sqrt{ {2 + 4 \cos \theta + 2 \cos^2 \theta \over 2}},$

$\rho = \sqrt{ 1 + 2 \cos \theta + \cos^2 \theta} = 1 + \cos \theta,$

quod erat demonstrandum.

[edit] Area derivation

The objective is to integrate the area of the cardioid

r = 1 - cosθ

The integral is

$A = \int \int dA = \int_0^{2\pi} \int_0^{(1 - \cos \theta)} r \, dr \, d\theta$ .

Integration with respect to dr yields

$A = \int_0^{2\pi} \left[ {1 \over 2} r^2 \right]_0^{(1-\cos\theta)} \, d\theta$

$= \int_0^{2\pi} {1\over 2} (1 - \cos\theta)^2 \, d\theta$

$= \int_0^{2\pi} {1 \over 2} (1 - 2 \cos\theta + \cos^2 \theta) \, d\theta.$

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

$A = {1 \over 2} \left\{ [\theta]_0^{2\pi} - 2[\sin\theta]_0^{2\pi} + \int_0^{2\pi} \cos^2 \theta \, d\theta \right\}.$

The second term vanishes, and integrating the third term yields

$A = {1 \over 2} \left\{ 2\pi + \left[ {1\over 2}\theta +{1\over 4}\sin 2\theta \right]_0^{2\pi} \right\}$

$= \pi + {1\over 2} \left[\pi + {1\over 4} (\sin 4\pi - \sin 0)\right].$

The last term within brackets vanishes, so that

$A = \pi + {1 \over 2}\pi = {3 \over 2}\pi.$

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)

Retrieved from "http://en.wikipedia.org../../../c/a/r/Cardioid_proofs.html"

Categories: Article proofs | WikiProject Mathematics