Cosine and Sine absolutely rule all straight and arc forms of spatial and opposing force purposeful/displace-meants;

and even permeate deeper into our world as also find same spatial and force displacements in waveforms of light, heat, electricity/magnetism, sound etc. all the same.

.

The Ancients handed these down over the milleniums to describe the consistently same, minimal definition, pivotal, forces found in their known basic known devices so well, even holds together in the waveforms they could perhaps not define, beyond their nagging senses except for perhaps water waves. These were about the same lines of force in all, irregardless of the transferring devices, that turns out can even be air itself as mute testimony to how far these things go across the board.

.

As in all other basic examples they found cosine/sine to rule in support work against loadings etc.

rock as a non-malleable example

metal as a malleable example

rope/fiber as a flexible example

wood as a stiffened fiber of at times both malleable and flexible in ranges type hybrid

>>that could also be carved and used for fuel unlike the malleable/non-malleables.

.

Rope work is just one lesson of lessons in these things, as a leading example of the flexibles category/family/class.

Flexibles are uniquely different to rest of classes in that:

>>doesn't resist on the cross axis/ only the inline axis

>>and then only in the tension/not compression direction on that given inline axis

>>is in it's malleable/form-able context at room temperature w/o hammer, drill, nor saw

>>only takes it's forged rigid form in loading, and only to the limits of that loading

.

Tangent is a very useful pre-configured shortcut; simply sine divided by cosine

>>so if multiply times a cosine value (load force), the cosine value drops out as x/x=1 for multiplier, leaving the sine value(side force) only, thus revealed.

.

.

This saves from going from load force to calc the line tension by division of cosine, to then multiply by sine to get side force

>>So can be very handy, as saves a step, AND more likely to know the load, than angled line tension...

>>Tangent, also, can be plotted on the clock fairly accurately to the common 45degrees point (7.5clock mins)

>>most Naturally accurate again, in most used range first 5 minutes on clock; 12-1

(BUT does not have the flip to reveal other half of 45-90 degrees as sine and cosine oblige to so well)

.

.

Most system and device fails from destabilizing influences of side/sine cutting across co(lumn)sine of required support against loadings.

In compression , side force value direction is asserted outwards from center

>>buckling from this to far outwards propulsion compromising column.sine(cosine), is easier to see .

>>and most expressed in center, as most leveraged from each end.

But, also, can have tension buckling, harder to even visualize as side force in tension pulls towards more correctly inline.

>>perhaps at some point too much as 2 force lines can 'slide' past each other (?)

>>to separate but in perhaps curiously in a half of a water meniscus shape AND angles

(effect exemplified in popular rigids under tension with lateral slider joint in between)

.

In both compression and tension directions on axis,

seem to have and 'elastic model' of response(how i keep them straight) for the side force directions:

>>in compression side forces swell outwards as ends compress in (like would expect compressing foam pillar)

>>in tension side forces pulls inwards as ends pull outward (like would expect stretching foam pillar)

But, in each case, failure seems to come from destabilizing lateral/cross forces more so, than outright longitudinal/inline causes.

.

These numbers decode the cryptic patterns, sifting to the cosine and sine piles.

edit/forgot:

min.deg.rule tangent

0 0 00 0.0000000000

1 6 11 0.1051042353

2 12 22 0.2125565617

3 18 33 0.3249196962

4 24 44 0.4452286853

5 30 57 0.5773502692 plus 5 %

6 36 72 0.7265425280 plus 10 %

7 42 91 0.9004040443 plus 20 %

m 45 100 1.0000000000 memorize