What is the Function of sin(x)?

[Helicobacter]

I searched my textbook and Wikipedia-D but I couln't find the function of sine composed with operations and values.

sin(x)=what?

I typed it in my graphing calculator but I can't really figure out the formula by only looking at the outputs. What operations does the calculator execute with my input x to spit out the output?

Thanks in advance!

 

[topsquark]

I searched my textbook and Wikipedia-D but I couln't find the function of sine composed with operations and values.

sin(x)=what?

I typed it in my graphing calculator but I can't really figure out the formula by only looking at the outputs. What operations does the calculator execute with my input x to spit out the output?

Thanks in advance!

Is this what you are looking for?
sin(x)= \sum_{n=0}^{ \infty} \frac{(-1)^{2n}}{(2n+1)!} x^{2n+1}

Otherwise you have to go back to Trigonometry and define the sine function in terms of a right triangle. There is no "closed-form" (aka "nice looking") function for the sine function.

-Dan

 

[HallsofIvy]

In other words, sin(x) is not an "algebraic" function.

 

[shaner-baner]

I know there are tons of ways to approximate a sine function, the most obvious being taylor series and numerical solutions of y''+y=0, but does anyone know how the "average" scientific calculator does it? The "best" way I think is to first map the argument onto (0,2pi) and then take advantage of symmetry to make your interval (0,pi/2). Taking pi/2 as a worst case value, and using the macluauren series I needed to go up to x^19 to get a value "equal" to 1 within double floating point accuracy. Is this what a calculator does? My best guess is that calculates on a smaller interval than (0,pi/2) where the series converges faster and then uses various identities to go back up.
Of course I don't think taylor series are your only option. There are things like accelerated series. I believe brent and soloman have done work in quadratically convergent methods for elementary functions (exp,sin,cos,log). (R. P. Brent, Fast multiple-precision evaluation of elementary functions, J. ACM 23 (1976) 242-251)

 

[Nimz]

I recall reading an article in College Math Journal, a publication of the MAA, that showed a very fast way to get sin x, most likely the method used by calculators. After a little bit of searching, I found the article was in the November 2001 issue: College Math Journal: Volume 32, Number 5, Pages: 330-333.

CORDIC: Elementary Function Computation Using Recursive Sequences
Neil Eklund
Using a single family of recursion relations, it is possible to calculate products, quotients, sines, cosines, arctangents, square roots, hyperbolic sines and cosines, logarithms, exponentials, and hyperbolic arctangents. That’s the way computers do it.

I have misplaced all my CMJ's, so that's about all I can dig up on it atm, besides the official CMJ website: http://www.maa.org/pubs/cmj.html

 

[shaner-baner]

After looking on the internet a little bit I found the so called cordic method. It's been around for 40 years or so and was used on the first hand calculators. Check out the website: www.emesystems.com/BS2mathC.htm