How Were Sine and Cosine Discovered?

[eax]

There is a symbol that looks like this http://mathworld.wolfram.com/nimg268.gif

The symbol is like a line that is curved on both ends.

Thanks in advance!

 

[Muzza]

It's an integral sign.

 

[jetoso]

The natural logarithm of x is equal to the integral of (1/t)dt from 1 to x.

 

[eax]

I don't think I'm going the right way with my problem. Anyway out of interest I want to know how to calculate the sin(x) without using the function(or cos, tan etc.) on the calculator.

 

[Icebreaker]

On the unit circle, there are places where sin(x) is defined exactly. Other than that, you could also use linear approximation, but that would require the derivative, and thus cosine.

A more precise way would be:

http://mathforum.org/library/drmath/view/64635.html

 

[Integral]

Use a Taylors series.

$$\sin (x) = x - \frac {x^3} {3!} + \frac {x^5} {5 !}- \frac {x^7} {7!}+ ...$$

The more terms you use the more accurate the result. This is the math behind small angle approximations. For small angles sin(x) ~ x.


x must be in radians.

Edited per Mathwonks correction.
OPPS!

 

[HallsofIvy]

I am told that calculators use the "Cordic" algorithm for things like sin, cos, log, exp.

Apparently it is faster than the Taylor's series. Here is a link to a website about it:
http://www.dspguru.com/info/faqs/cordic.htm

My father used to call the integral sign a "seahorse"!

 

[Icebreaker]

What exactly is the difference between an equal sign and an equivalence sign (as used in first post)?

 

[selfAdjoint]

What exactly is the difference between an equal sign and an equivalence sign (as used in first post)?

In the case of the OP's image, it means that the natural log is DEFINED by that integral. This is where ln, and e, and all come from, integrating the hyperbolic function 1/x.

 

[mathwonk]

in Integra's post there are sign errors, i.e. the signs should alternate, so the series given will not compute sin(x) at all.

 

[eax]

these are to get approximates What is the original sine formula?

 

[dextercioby]

What do you mean...?That formula converges for every argument and can definitely e put under a simpler form,using the summation symbol "sigma".

Daniel.

 

[Zurtex]

these are to get approximates What is the original sine formula?

$$\text{Sine}(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$$

 

[chroot]

these are to get approximates What is the original sine formula?

There's no such thing as an "original sine function." The sine function is transcendental, so it cannot be represented exactly in any algebraic form of finite length.

- Warren

 

[dextercioby]

Yes,Mathwonk,i'm sure everyone noticed that

Daniel.

 

[eax]

So then how was sine discovered? Is it possible to get sin(x) without using an infinite series equation?

 

[dextercioby]

You've been answered to the last question already.
As for the first,i can't trace it back earlier than Claudius Ptolemeu's tables on light refraction in water.

Daniel.

 

[Crosson]

Sine tables were constructed empiricaly, to a decent number of significant digits, in the old days.

How is that for rigor!

 

[dextercioby]

And what is that supposed to mean...?You know,a little is always better than nothing...


Daniel.

 

[mathwonk]

One remark about defining sin, which may be obvious, is: if you look at the circle function definition, the sine function is the inverse of the circular arclength function.

i.e. the possibly more natural function is the function arcsin(y) taking y to the arclength along the circle from the point (1,0) where the circle meets the x axis, to the point on the circle at height y, for 0 <= y <= 1.

this function takes values from 0 to <pi>/2, and sin is its inverse on that interval.

this suggests the definition of sin as the inverse of the arclength integral

i.e. of the integral of dt/sqrt(1-t^2) from t=0 to t=x.

this is the analog of defining ln(x) as an integral, and then defining e^x as its inverse, or of defining an elliptic function as the inverse of the integral of dt/sqrt(1-t^4), as Euler did.

My opinion is it is natural to wonder how to express circular arclength as a function of some simpler parameter such as height, but rather less natural to ask about the sin function, its inverse.

 

[BobG]

So then how was sine discovered? Is it possible to get sin(x) without using an infinite series equation?

Crosson's right. The sine and cosine were discovered empirically maybe as much as 3000 years before calculus, integration, Taylor, or McLaurin series were invented. They were simply using the concept of similar triangles. If the angles of one triangle were equal to that of a second, then the sides of both triangles must have the same proportions.

In fact, they eventually led to Euler's logarithmic functions. Trig tables were used for multiplication and division via your cosine/sine addition/subtraction identities for quite awhile before logarithms were ever invented. Napier drew from that and came up with base 10 logarithms, and Euler expanded on Napier's work to get the natural logarithms and e.