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Why is this the case?

- Thu, 25 Apr 2019 01:23:27 EST bUpRMfRf No.79388
File: 1556169807858.jpg -(63514B / 62.03KB, 700x538) Thumbnail displayed, click image for full size. Why is this the case?
I've been enrolled in a Trigonometry course this semester and I have been enjoying it. I also think the main Pythagorean theorem is one of my new favorite equations, but I just don't understand one thing. Why is the result of the expression equal to one?

sin²β + cos²β = 1

We've learned about "proves" where you try to arrive to the answer by choosing the busiest side and in this case the left side, but how would you go about proving this specific this? I heard that more advanced math knowledge was needed (e.g. Calculus) but is that really true or can this be proven using what is learned in Trigonometry itself?

Also, in other Trig courses, do instructors actually assign this expression as a homework question all to itself? I thought this was just a formula given to help solve problems and not an actual problem.

I also recently learned that Pythagoras was a philosopher and even had his own cult. What are some of the other things this man was known for?
Phoebe Ducklefuck - Fri, 26 Apr 2019 08:00:01 EST FyzhtNRJ No.79395 Reply
1556280001747.png -(26715B / 26.09KB, 1090x1155) Thumbnail displayed, click image for full size.

Proofs are rewards that keep giving. They aren't exercises to help you remember how to apply a formula, but exercises to help you grok it.

The cult was more of a religious order, a retreat of monks living aesthetic lifestyles, vowing silence, keeping to themselves and meditating on understanding mathematics, the true perfection of the universe, the code that the gods themselves wrote existence with.

They held that matter was formed of geometries, and that the dodecahedron must be kept secret from the masses as it was a regular prism beyond comprehension that caused problems with the four elements of Aristotle lining up with known regular prisms of the time.

pic tangentially related
Nicholas Packleman - Fri, 26 Apr 2019 09:24:10 EST QiRUncgI No.79396 Reply
I know this is a spergy thing to make a whole post over, but the monks lived ascetic lifestyles. Although some people would certainly find that aesthetically pleasing.

You're right about everything else.

That is all.
Phoebe Ducklefuck - Fri, 26 Apr 2019 12:43:49 EST FyzhtNRJ No.79398 Reply
1556297029747.png -(168936B / 164.98KB, 500x375) Thumbnail displayed, click image for full size.

Hahaha, fuck autocorrect.

Yeah I know the difference, thanks Nick Spackleman
press - Fri, 26 Apr 2019 16:11:50 EST kphkeya5 No.79399 Reply
his hatred of beans (or legumes in general?) which according to some most likely bullshit stories lead to his death as he was fleeing an aggressor yet refused to enter a field of beans in order to escape. makes you wonder what kinda stuff him and his disciples did that might have lead to disdain for beans

also, you dont need calculus to prove most trig identities. by euclides definition the only tools needed for geometry are a circle (ie a peice of string) and a ruler but obviously that is stretching it. since calculus has only been around for a few hundred years you could deduce trig identities with other reasoning, calc is just much more convinient in most cases.
cant give a properly formulated example of the top of my hat, sorry. id have to search for one

man, geometrics is fun though, especially since it developes your spatial ( or rather geometric ) understanding. y'all ever been told that humans cant imagine a four dimensional space? i betcha thats because some people think the fourth dimension in spatial imaging has to mean time, i mean its pretty fucking easy to imagine a scalar field that starts in R³. time is pretty fucked up though

also; KIRT Latex support when?
George Greenwell - Sat, 27 Apr 2019 19:39:21 EST 0rxAM/dP No.79401 Reply
>sin²β + cos²β = 1
>but how would you go about proving this specific this?

Legend: CO - Catetus opposite (to β); CA - Catetus adjacent (to β); H - hypothenuse
Pythagorean theorem says CO^2 + CA^2 = H^2
Divide both sides by H^2: (CO^2 + CA^2)/H^2 = 1
Reorganize: (CO/H)^2 + (CA/H)^2 = 1
But Catetus opposite to β over hypothenuse is by definition the sine of β,
and Catetus adjacent to β over hypothenuse is by definition the cosine of β,
so we have: sin^2 β + cos^2 β = 1
As we wanted to show.
Beatrice Lightville - Mon, 29 Apr 2019 22:08:17 EST bUpRMfRf No.79404 Reply
> The cult was more of a religious order, a retreat of monks living aesthetic lifestyles, vowing silence, keeping to themselves and meditating on understanding mathematics, the true perfection of the universe, the code that the gods themselves wrote existence with.

That sounds like a really unique perspective on the world and/or universe. It makes me wonder if they would have believed the world is a simulation with mathematics being the programming language it is coded in.
Cyril Poblingpick - Tue, 30 Apr 2019 07:04:54 EST FXRPIW2b No.79405 Reply

That is how the world is. Numbers are real. Mathematica is true. The world you experience using your senses is simple phenomena with no inherent importance of meaning.
James Fuffingtodge - Tue, 30 Apr 2019 13:36:28 EST QiRUncgI No.79406 Reply
>>It makes me wonder if they would have believed the world is a simulation
Not on their own. They would have no frame of reference for the idea of a computer simulation. If the ancients did have the idea of some sort of disembodied mind that ruled the universe through invisible mathematical laws, they would just call it god, and in fact many did.
Nathaniel Darryhall - Wed, 01 May 2019 14:52:22 EST htsdNNzt No.79407 Reply
It means the legs of the triangle. Any side that is not the hypotenuse is a cathetus. Literally it is "vertical" or "perpendicular in Greek, for the two sides that meet at the right angle.
press - Fri, 03 May 2019 16:09:04 EST kphkeya5 No.79409 Reply
wouldnt you say the allegory of the cave loosely fits the idea of simulation and how we might perceive it as reality?
Betsy Boddleshaw - Fri, 03 May 2019 16:45:56 EST QiRUncgI No.79410 Reply
The allegory of the cave is a metaphor for epistemology, I don't think Plato was positing it as a realistic scenario, whereas the simulation argument might be taken for a metaphor about epistemology but primarily positions itself as a realistic explanation for reality. (cf. the Evil Genius argument; Descartes didn't really believe he was a brain in a vat, he was trying to prove a point.)
There really isn't a difference between the way our brain simulates the world and a computer simulation of the world; but that itself, comparing the brain to a computer, is just another metaphor, and largely the product of the repeating phenomenon in the history of science where the rest of the universe is re-contextualized as being 'like' whatever the most important technological advance of the day is (this has happened with everything from fire, writing, and wheels to rail-roads and electricity.)

Not to say that he or Pythagoras couldn't eventually understand the concept of a computer and programming languages, and be able to relate them to how the mind and possibly the universe works, but you would need to educate them about all the stuff that underlies those ideas first. Like, the number zero and real algebra would be the first big things.
Rebecca Sondlestock - Thu, 09 May 2019 09:36:08 EST fBrSG5+b No.79416 Reply
1557408968273.jpg -(106274B / 103.78KB, 786x1017) Thumbnail displayed, click image for full size.

And yet the work of flatland and explanations of dimensions beyond human comprehension are adequately expressed using the allegory of the cave.

I do enjoy that you make the distinction that belief is not relevant to argument. So rarely do people understand that critical nuance.
!scyTheNg3k - Thu, 30 May 2019 18:10:02 EST +fcpEO88 No.79430 Reply
the proof was given correctly above but the dude used latin for some reason.

sine is the ratio between opposite side and hypotenuse. sin(x) = O/H
cosine is the ratio between adjacent side and hypotenuse. cos(x) = A/H

pythagorean theorem: (adjacent side)^2 + (opposite side)^2 = (hypotenuse)^2
O^2 + A^2 = H^2
divide both sides by H^2
O^2/H^2 + A^2/H^2 = 1
(O/H)^2 + (A/H)^2 = 1
by definition
sin(x)^2 + cos(x)^2 = 1

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