Trigonometric functions

where are the trigonometric functions derived from, how the hell did someone think of creating sin/tan/cos (arc funcs too)

thanks, non pic related

Other urls found in this thread:

businessinsider.com/7-gifs-trigonometry-sine-cosine-2013-5
en.wikipedia.org/wiki/Hipparchus#Geometry.2C_trigonometry.2C_and_other_mathematical_techniques
en.wikipedia.org/wiki/Bessel_function
en.wikipedia.org/wiki/Beta_function
en.wikipedia.org/wiki/Airy_function
en.wikipedia.org/wiki/Catenary
twitter.com/SFWRedditGifs

I don't know the history of trigo functions, but I can tell you that , nowadays, the formal definitions of sine, cosine, tan... are infinte sums based on the Taylor series of the functions (aka sum of polynomials).
Sine in spanish is "seno" which means "boob" and the greek letter "theta" pronounced in spanish is "teta" that also means "boob" so, if you say "sine of theta" you are saying "boob of boob". wew.

Speaking of which, since the question's answered, are there any geniuses here who wouldn't mind getting paid to take care of some work for me?

sorry for being a maths noob but are the sin/cos/tan functions derived from the taylor series then? if not where from, where do they come from, how where they founded

They come from classical geometry, and the motivation, like most of math, comes from practical problems. When building things like pyramids and towers, a lot of simple geometry comes into play. They needed to be able to estimate the sizes of things, and they discovered that there were relationships between the edge lengths of triangles based upon the angles between any given two.

You have to remember that the math used by these people does not resemble anything we do today. It was very visual, and a lot of work was done using some old notion of ratio. With these few things in mind, it seems natural to think of things such as sine, cosine, and tangent. All of those proofs you've seen have geometric counterparts.

The beauty of modern math is that it allows us to extend our thoughts farther than our intuition alone allows. Once more modern mathematicians found suitable non-geometric representations of the ideas of sine and cosine, we were able to produce all kinds of things that are probably ridiculously difficult, if not impossible to see geometrically.

poster here.. I reread this.. Dear lord my English is atrocious here. Anyway, the idea still holds. I asked this same question in a calculus course while still in high school. Fortunately I had a retired uni professor that taught for fun at the local community college, and he gave me the gist of why we use the "crazy" formulas we do now.. It's because they are equivalent to the other ideas, and allow us to dig much deeper than the original Greek diagrams.

The only way for you to truly see this is to find older material referencing the classical usage of sine, cosine, etc., and then try to find proofs of the equivalence. Once you're convinced that they're the same, you have to ask yourself why you should even care that there is another way of representing them. This is a big question in all of math: why should I care? Well, you already have the answer to that. Using euler's identity and the infinite series representation you can prove almost any trig identity with relative ease. Go ahead and try to prove the identities geometrically and see how long it takes.

Even better: How do you tell a computer to handle trigonometric material if you represent it using diagrams?

This is my last post, then I have to go to class.
One important historical note: the names sine and cosine are not the original terms used to discuss things about the triangle. I am going off of the top of my head here, so don't take my word as gold. If I am not mistaken, the original terms to describe the relationships of the sides of a triangle were something else, and the words we use today are just a historical accident.. Mistranslations from arabic and this and that. History is fun!

**I'm speaking metaphorically here, so further would have been the correct choice of word**

"extend our thoughts further..."

but where do they come from, its obvious sin isn't a constant value. When programming a calculator how do you tell it what sin is or cos or tan. Where did the values come from

businessinsider.com/7-gifs-trigonometry-sine-cosine-2013-5

nice

Original sin was given to mankind by God, later cosplay came into fashion.

Kek still haven't got a good answer, I might have to ask reddit ;(

It's just a made up ratio. For every triangle people noticed that these ratios were always the same so made tables of it that you could consult whenever you wanted to work out a triangle. Then we started calling it a function and found ways to use infinite series to calculate them instead of actually measuring triangles and tabulating it. Wildberger is right it is a fudge.

They're derived from polar coordinate systems. This is why they repeat values every 2pi.

sure why not :^)

read a history of math book

en.wikipedia.org/wiki/Hipparchus#Geometry.2C_trigonometry.2C_and_other_mathematical_techniques

They are transcendental functions derived from forcing a non-linear concept (rotation) to be linear (angles).
The only rational way is to use rotor coordinates and actually be able to get a number from the function without a calculator

>haven't got a good answer?

You received the basic historical development in a nutshell you stupid fucking faggot.

>but, like, how they get muh numberz when I ask sin(pi/3)?

Your questions is basically why is 3/4 = .75?
Why is sqrt(2) = 1.41......
It's fucking retarded
This trig functions represent an idea: a ratio between sides of a triangle. People noticed that if you indeed have a triangle, then these ratios, no matter what the edge lengths are, are going to be fixed. God fucking damn you stupid mother fucker. Go post on reddit.

Sin and cos are derived from exponential function:
[eqn]e^x=exp(x)=\sum_{n=1}^{\infty}{\frac{x^n}{n!}}[/eqn]

for [eqn]x=z*i[/eqn] you will get [eqn]cos(z)=Re(exp(z))[/eqn] and [eqn]sin(z)=Im(exp(z))[/eqn]

This also provides a way to calculate sin and cos.
You can separate the imaginary and real part of the exponential function very easily.

btw:
[eqn]tan(z)=\frac{sin(z)}{cos(z)}[/eqn]

thank you, most satisfactory answer I have received in my life, tyvm TEACHER ;)

This is the right answer.

Like every special function in maths, their form (most likely as a power series) appeared often, so it became convenient to name them.
Note that we can define, for example sin, in multiple ways - as a power series, as a ratio, as a solution to an ordinary differential equation, etc.
There are a lot of other special functions in maths, such as:
- Bessel functions: en.wikipedia.org/wiki/Bessel_function
- Beta functions: en.wikipedia.org/wiki/Beta_function
- Airy functions: en.wikipedia.org/wiki/Airy_function

With this context, you shouldn't see the trigonmetric or cyclometric functions (yes, another name for the inverse trig functions) as anything scary, since there are a lot more out there. These sine and cosine functions really are nice special functions.

High-school tryhard fresh out of Calc I that can't even [math]\sin \cos \exp[/math] properly detected.

Take a unit circle and sweep out an angle of [math]\theta[/math] along the edge. The [math]x[/math] and [math]y[/math] coordinates of the point your finger stops at are [math]\cos\theta[/math] and [math]\sin\theta[/math], respectively.

[math]\tan\theta[/math] is just a convenient way of writing [math]\dfrac{\sin\theta}{\cos\theta}[/math], since it comes up so often.

The arc functions are the inverse of this process. For example, [math]\arccos x[/math] asks the question "what angle did I have to sweep out along the edge of the unit circle to get to an x-coordinate of [math]x[/math]?"

>capital C
>capital S
>not TikZ
pls stahp

...

This.

OP the sin, cos, and tan functions exist as a set of unique functions which satisfy unique properties. They were simply discovered. But historically I would believe .

I'm just saying that you don't have to think of them as functions reliant on a triangle anymore, though it is helpful. You can think of them as the unique functions which satisfy certain properties defined by way of Euler's formula and just use them like that.

what the fuck is the point of hyperbolic trig functions?

The purpose of those is to simply be a shorthand for functions which appear extremely commonly in application, mostly for engineers.

For example, when calculating the height of any dangling bridge, the shape of the bridge resembles a hyperbolic function. Not, as many people think, a hyperbola like x^2.

Our calculus class back when I was in high school had us calculate the heights of bridges and of beams holding bridges up back when I was in high school so I remember this.

It's just a shorthand for an extremely useful set of functions that happen to appear often in applications.

>the shape of the bridge resembles a hyperbolic function
This is correct: en.wikipedia.org/wiki/Catenary

That was it. The catenary.
Very fun application.

>High-school tryhard fresh out of Calc I that can't even sincosexp properly detected.
???
I dont think there is anything wrong with what I said and unlike your explanation I gave a reason how the functions are derived and how you can calculate its values.

Also (because you dont seem to be aware of it) what I said and what you said are very similar. The real and imaginary parts of the exponential function describe:
>The x and y coordinates of the point your finger stops at are cosθ and sinθ, respectively.
But on the complex plane.

Similar to sweeping out an angle on the unit circle, but this time the important part is the total area you've swept out between the unit hyperbola and a straight line from your finger to the origin (i.e. the yellow area in the diagram). The x and y coordinates resulting from a swept area of u are [math]\cosh u[/math] and [math]\sinh u[/math], respectively.

And you expect someone who doesn't understand trig functions to begin with to know this? Like most on Veeky Forums you were looking to impress, not to teach.

I thought it was a good way to show how different things in math relate to each other.
And anyone who finished Highschool math did probably understand most of it.

Also both OP and seemed to be aware of what the purpose of these functions are but not how they are derived, properly defined and relate to other things.

Btw "showing off" on an anonymous image board seems to be a pretty silly idea.

Actually it's even more similar to the unit circle thing, since sweeping an angle of [math]\theta[/math] out on the unit circle results in an area of [math]\theta[/math] when a similar scheme as in that diagram is adopted.

pretty much everyone understands trig funcs u fgt