## General Math

### Pentagon

2015-01-02

A regular pentagon has the following dimensions: AB=BC=CD=DE=AE=BG=EG=1 Height AH=EK=$\frac12\cos 18 = \frac12\sin 72= \frac12\sqrt{5+2\sqrt5}$ AC=AD=BD=BE=CE=Φ Circumcenter AJ=$\sqrt{\frac{5+\sqrt5}{10}}$=BJ=CJ=DJ=EJ BF=EF=Φ/2 = cos 36 = sin 54 = $\frac{1+\sqrt5}{4}$ Inradius JK=HJ=$\sqrt{\frac{5+2\sqrt5}{20}}$=$\frac{AH}{\sqrt5}$ CG=DG= Φ-1 = 1/Φ = $\frac{\sqrt5-1}{2}$ FH= cos 18 =sin 72 = $\sqrt{\frac{5+\sqrt5}{8}}$ AF=FG= cos 54 …Read the Rest

### Factoring numbers with square root terms

2013-07-30

Quite often, when doing calculations on polyhedra, you will find yourself with complex equations like $\frac{20+7\sqrt3}{1+2\sqrt3}$.

Is the top evenly divisible by the bottom? I certainly can’t tell just by looking at them.

There must be a method to factor the numerator.

### sqr(A+B*sqr(C))

2013-07-27

Lately, I have been doing a lot of math involving square roots of numbers added to square roots, in the form of $\sqrt{A+B\sqrt{C}}$, this is called a “nested radical.” Normally, you would not be able to simplify any further, unless there was a common factor …Read the Rest

### Law of Sines and Cosines

2011-11-07

The law of cosines relates the sides and angles of a triangle. $a^2=b^2+c^2-2bc\cdot \cos\alpha \\ b^2=a^2+c^2-2ac\cdot \cos\beta \\ c^2=a^2+b^2-2ab\cdot \cos\gamma$ It can also be rearranged to: $\large\alpha=\arccos\left(\frac{b^2+c^2-a^2}{2bc}\right) \\ \large\beta=\arccos\left(\frac{a^2+c^2-b^2}{2ac}\right) \\ \large\gamma=\arccos\left(\frac{a^2+b^2-c^2}{2ab}\right)$ As long as all three sides or at least one side and two angles …Read the Rest

### Phi, the Golden Ratio

2011-10-23

Phi $(\Phi, \phi)$ is a Greek letter that mathematicians have assigned to a specific ratio or proportion, called the golden ratio, that most people find to be attractive in art, architecture, and nature. The golden ratio is illustrated as $\frac{A}{B}=\frac{A+B}{A}\equiv\phi$. The only positive solution is …Read the Rest

### Table of exact trigonometric functions

2011-10-19

Since I noticed I had to keep looking some of these up, I place them here, just for reference, gathered from around the internet. Many of these formulas can be written in different ways, but I have simplified them as much as possible. Where, $\phi=\frac{1+\sqrt{5}}{2}=1.6180339887498948482045868\ldots$, …Read the Rest

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