An Approach to the Triangle

In this Section , we look at { Triangles with Interger Sides } ,

• where the values of { Sine-Square } & { Cosine-Square } for all 3 angles are always { whole-fractions } .

Let us now take a quick look at the { cosine-square } & { sine-square } functions :

• And we simply note here that both functions are monotonic over the range of { 0 to 90 degrees } :
  • so that the { sine-square } function can be used as a ranking function of { } over this range , and that is to say :
    • if the value of [ sine-square of { X } ] is greater than the value of [ sine-square of { Y } ] ,

      then { X } is greater than { Y } .

  But more on this later in the last [ Topic ] of this Section .

Let us now bring in this { trigonometric identity } :

Let us now bring back this formula for the [ Area-of-the-Triangle ] , { A-O-T } , from [ Section II - Topic #4 ] :

And because [ area = one-half * base * height ] , we can express :

We then have :

yielding :

And { Cosine-Square } may be found via :

Thus , as we have previously stated in [ Section II ] :

• For [ Triangles with { Integer Sides } ] :

  the values of { Sine-Square } & { Cosine-Square } are always { whole-fractions } for all of the { 3 angles } .

And we shall look at this next .

Let us now take a quick look at 3 examples of [ Triangles with { Integer Sides } ] :

And we summarize the calculations as follows :

L1	L2	L3	16 * A-O-T	sin^2	sin^2	sin^2	cos^2	cos^2	cos^2
1	1	1	3	3 / 4	3 / 4	3 / 4	1 / 4	1 / 4	1 / 4
17	19	23	402,675	402,675 / 763,876	402,675 / 611,524	402,675 / 417,316	361,201 / 763,876	208,849 / 611,524	14,641 / 417,316
11	13	17	81,795	81,795 / 195,384	81,795 / 139,876	81,795 / 81,796	133,589 / 195,384	58,081 / 139,876	1 / 81,796

Let us now pick the { 17 : 19 : 23 } Triangle , ( i.e. the diagram in-the-middle ) , and ask this question :

• Can we find another [ Triangle with { Integer-Sides } ] so that the value of [ Sine-Square ] is [ 402,675 / 763,876 ] ?

But before we answer that , let us take a quick look at merging [ Right-Angle Triangles with { Integer Sides } ] , next .

Let us now attempt to merge 2 { Right-Angle Triangles with Integer Sides } , the { 5-12-13 Triangle } & the { 3-4-5 Triangle } :

And we have the { merged triangle } as per the the diagram on-the-right , above .

We then have this table below for the value of [ Sine-Square ] , for the diagrams on-the-left & on-the-right , above :

L1	L2	L3	16 * A-O-T	sin^2	cos^2
5	12	13	14,400	14,400 / 97,344	82,944 / 97,344
25	52	63	6,350,400	6,350,400 / 42,928,704	36,578,304 / 42,928,704

As expected , both fractions : [ 14,400 / 97,344 ] & [ 6,350,400 / 42,928,704 ] , reduce to [ 25 / 169 ] .

This then proves that the same value of [ Sine-Square ] can arise from more-than-one [ Triangles with { Integer Sides } ] .

• But what about the last question in [ Topic #4 ] above ???

Let us suppose , for the time being , that we are restricted to using only { positive integers less than [ 16 ] } for our [ Triangles with { Integer Sides } ] .

• We can then construct a finite-set / limited-number of many [ Triangles with { Integer Sides } ] , using the different combinations of { integers } , with :
  • the { angle } being differently sized , and
  • the corresponding values for { Sine-Square } also being different .

  And we can then rank these [ Triangles with { Integer Sides } ] in accordance to the associated values of { Sine-Square } , in an ascending-order .

  • so that the associated sizes of { } are also ranked in an ascending order .

Let us now draw two (2) random lines on a piece of paper that do not intersect at { right-angles } ,

• and attempt to measure the { intersection angle } that is less than [ 90 degrees ] , denoted by { angle } .

We can then do a physical comparison with our { ranked series } created above and identify two (2) adjacent sizes of { } , i.e. :

• { } & { } respectively ,

such that :

And we have the corresponding relation :

We have therefore developed a system of closing-in on the size of { angle } .

And , if we needed more accuracy , we can always relax our first restriction above by :

• using { positive integers less than [ 32 ] } ,
• using { positive integers less than [ 64 ] } ,
• using { positive integers less than [ 128 ] } ,
• and so-on-and-so-forth .

  to close-in on the gap between { } & { } .

Our open question here is this :

• The { 3 angles } of a triangle always add up to [ 180 degrees ] , so that each { Triangle with Integer Sides } does specify a particular relation between 3 { whole-fractions } ,
  • namely the values of { Sine-Square } for the 3 angles .

  Does this have anything to do with anything at all ??? ( Again , reader's call ) .