# What does this symbol mean?

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

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

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

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
Staff Emeritus
Gold Member
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.

Edited per Mathwonks correction.
OPPS!

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HallsofIvy
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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)?

Staff Emeritus
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Dearly Missed
Icebreaker said:
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
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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.

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

dextercioby
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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
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eax said:
these are to get approximates What is the original sine formula?
$$\text{Sine}(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$$

chroot
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eax said:
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
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Yes,Mathwonk,i'm sure everyone noticed that Daniel.

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

dextercioby
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As for the first,i can't trace it back earlier than Claudius Ptolemeu's tables on light refraction in water.

Daniel.

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

How is that for rigor!

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

Daniel.

mathwonk
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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.

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BobG