What's the sine/cosine of X?

Syth

Inspired by the "Whats the square root of X?" thread, i thought about sine (and conversally cosine). Sine can only be represented by a non-terminating polynomial, he well know Maclarian series. Makes some kind of sense. But if you can't calculate what the sine of X is, does it really exist? Can we really use it in formula?

But what about triangles? Like most people I was first introduced to trigonometric function via angles and triangles. Is there no way of converting simple triangles into algebric functions that result in a polynomial function for sine?

ecksor

Originally posted by Syth
Sine can only be represented by a non-terminating polynomial, he well know Maclarian series. Makes some kind of sense. But if you can't calculate what the sine of X is, does it really exist?

It exists as the ratio between two sides of a right angled triangle in the same way that PI can be defined as a ratio, or does that raise the same question for you?

Can we really use it in formula?

I don't see why not. We have a very clear idea of what it represents and how to manipulate it. The difficulty you mention is just about how to represent it numerically as far as I can make out.

But what about triangles? Like most people I was first introduced to trigonometric function via angles and triangles. Is there no way of converting simple triangles into algebric functions that result in a polynomial function for sine?

I think the best way of seeing a problem with that is to think about what happens when you try to get the derivative of that polynomial. If the number of terms in the polynomial was finite then you wouldn't be able to continually differentiate from sin to cos and back again indefinitely in the way that we can.

smiles

Well, if you're into complex analysis then another way of looking at cos and sin is in terms of the exponential, known as another of Euler's theorems

e^(iƟ) = 1 + (iƟ) + ((iƟ)^2)/2! + ((iƟ)^3)/3! + ((iƟ)^4)/4! + .... = cos(Ɵ) + isin(Ɵ)

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Capt'n Midnight

All I can see are squares ! (fonts?)

Looking at a clock and taking three O'Clock as 0 then then at twelve O'Clock the top of the hand is twice as high as it would be at two O'Clock (or nine)

If you express angles in radians instead of degrees, ie. measuring it in terms of the length of the circumference, then at very small angles the sine of the angle is equal to the angle.. Because it's a unit circle the hypotenous is of length one. - Since the angle is also very small that part of the circumference is nearly straight. Also the adjacent side is also very nearly length one. - so the sine of small angles becomes a problem of triangles. Large angles are simply lots of little ones together.

smiles

Originally posted by Capt'n Midnight
All I can see are squares ! (fonts?)

They're meant to be theta signs. :doh:

As a side note:

( pi^4 + pi^5) ^ (1/16) = e

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Capt'n Midnight

Phi don't work either

We can't compute PI exactly either - but 355/113 is an approximation good for to calculate the earth's circumference to within 3.4 meters (11 feet).

In the old days I suppose they used only the first few terms of the expansion to calculate sine (and hence cosines) for the log tables - then there was the whole fun of figuring out the maximum possible value for the remaing terms in the expansion to figure out your error range..

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Actually the original question is backwards.
If you have a right angled triangle then picking one of the other angles, then the sine of that angle is the length of the side opposite the angle divided by the length of the hippotomus (that saw the son in the square, Tallaght, etc. )

We KNOW what sine is - what is the angle ?

Any scientific calculator will give three different answers - degrees (Bablyonians), Gradians (French Revolutionaries) , and Radians (anyone ?)

BTW: Aliens would of course be familiar with Radians - doubtful they will other bases to the same degree. (wonders aloud - were the Bablyonians into equilaterial triangles since sixty degrees would have been a round number to them ?)

Syth

So if sine is so easily expressed as the ratio of sides of a triangle, why can't we write down the formala for sine in terms of simple arthimetic operations?

ecksor

Originally posted by Syth
So if sine is so easily expressed as the ratio of sides of a triangle, why can't we write down the formala for sine in terms of simple arthimetic operations?

Can you clarify that? sin(x) = opposite/hypotenuse seems simple enough to me, but I guess that's not what you want ...

Syth

Can you clarify that? sin(x) = opposite/hypotenuse seems simple enough to me, but I guess that's not what you want ...

Well you can't write it as a polynomial. I guess the real problem I have is that you can write it as a simple sum, but then you also can't. Does that mean that opposite/hypotenuse (and perhaps division in general) can be written as an infinte polynomial. It seems strange.

Capt'n Midnight

It's akin to squaring the circle
- there is no geometric way of calculating PI

You can draw a line and then a circle around it with the ratio of the two being PI , you can use the size of the line to make a square..

You can make an equilateral triangle easily and know the angles are all 60 degrees and that the sines are all root(2) (angle is PI/3 in radians) - similar things happen at other convenient angles - but PI still pops upthere