## 20454 Reputation

16 years, 63 days

## Another proof of stunningly beautiful id...

I propose a different proof of this remarkable identity (see  http://www.mapleprimes.com/posts/144499-Stunningly-Beautiful-Identity-Proved ) in which  directly constructed a polynomial, whose root is the value of LHS, and this is expressed in radicals.

For the proof, we need three simple identities with cubic roots (a, b, c -any real numbers):

Maple

It is known that the trigonometric functions of an integer number of degrees may be expressed by radicals if the number of degrees is divisible by 3. Simple code finds all these values ​​in the range 0 to 90 degrees:

[sin(`0`^`o`)=0,`   cos`(`0`^`o`)=1,`   tan`(`0`^`o`)=0,`   cot`(`0`^`o`)=infinity];

for n from 3 to 87 by 3 do

## Visualization of "Clock and Degrees" pro...

Clock:=proc(H, M)  # H and M - time in hours (0 <= H <= 23) and minutes (0<=M<60)

local A1,A2, A, B, B1, C1, C, E, F,alpha,t, T, T1, T2, P, G;

uses plottools, plots;

A1:=(irem(H,12)+M/60)*30;  A2:=M*6;

A:=circle([0,0], thickness=5);

P:=disk([0,0], 1, color=grey);

## Maple for "Clock and Degrees" problem...

Maple

Here are two procedures associated with a clock with hands. I think they are interesting not only for fun, but will be useful for teachers in schools in the preparation of such tasks.

The first procedure called  ClockHandsAngle  finds the angle between the hands of the clock at any time.

The second procedure called  TimeFromAngle , for a given angle finds all times in a given time range. The first procedure  is...

## Two sets of points in the plane...

Maple

Consider two sets in the Euclidean plane, each consisting of 4 points.

First set:  A(0, 0),  B(3, 4),  C(12, 4),  E(4, -1)

Second set:  F(0, -8),  G(12, -4),  H(9, -8),  K(4, -9)

It is easy to check that the set of all pairwise distances between the points of each of the given sets (6 numbers for each set ) are the same. At the same time it is obvious that there is no any...

 First 8 9 10 11 Page 10 of 11
﻿