# What does this symbol mean?

1. Mar 6, 2005

### eax

There is a symbol that looks like this image

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

2. Mar 6, 2005

3. Mar 6, 2005

### jetoso

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

4. Mar 6, 2005

### 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.

5. Mar 6, 2005

### 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

6. Mar 6, 2005

### Integral

Staff Emeritus
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!

Last edited: Mar 8, 2005
7. Mar 6, 2005

### HallsofIvy

Staff Emeritus
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"!

8. Mar 6, 2005

### Icebreaker

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

9. Mar 6, 2005

Staff Emeritus

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.

10. Mar 6, 2005

### 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.

11. Mar 7, 2005

### eax

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

12. Mar 7, 2005

### 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.

13. Mar 7, 2005

### Zurtex

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

14. Mar 7, 2005

### chroot

Staff Emeritus
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

15. Mar 7, 2005

### dextercioby

Yes,Mathwonk,i'm sure everyone noticed that

Daniel.

16. Mar 7, 2005

### eax

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

17. Mar 7, 2005

### dextercioby

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

Daniel.

18. Mar 7, 2005

### Crosson

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

How is that for rigor!

19. Mar 8, 2005

### dextercioby

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

Daniel.

20. Mar 8, 2005

### 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.

Last edited: Mar 8, 2005
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