Trying to figure out a single variable expression for Sin

1. Jun 6, 2012

mesa

Well, I am now trying to figure out a single variable expression for Sin. I have a couple ideas using some pieces of geometric formulas I have played with recently but this is still new to me. I'm not talking about sin x, but an algebraic expression for the sin wave.

Any thoughts?

Last edited: Jun 6, 2012
2. Jun 6, 2012

mathman

You need to clarify what you have in mind.

3. Jun 6, 2012

Number Nine

Do you mean that you want an expression for sin that involves only addition/subtraction, multiplication/division and exponentiation? Have you tried a Taylor polynomial?

4. Jun 6, 2012

HallsofIvy

Staff Emeritus
y= sin(x- ct) is a perfectly good formula for a sine wave moving with speed c. Did you have something else in mind?

5. Jun 6, 2012

mesa

An algebraic expression that will compute y for a given x on a sin wave.

6. Jun 6, 2012

mesa

Basically I want an algebraic expression (if it can be done in one piece) that will spit out the right y values for a given x
What is a Taylor polynomial?

7. Jun 6, 2012

Number Nine

Something that you'll learn about in calculus; it's a way of approximating certain kinds of functions using polynomials. The sin function is nice in the sense that it is actually everywhere equal to it's Taylor series.

8. Jun 6, 2012

HallsofIvy

Staff Emeritus
In terms of the Taylor series expression for x, about x= 0, also called the "McLaurin series", is $x- x^3/3!+ x^5/5!- \cdot\cdot\cdot+ ((-1)^n/(2n+1)!)x^{2n+1}+ \cdot\cdot\cdot$.

9. Jun 6, 2012

mesa

Is it possible to get an exact function for sin?

10. Jun 6, 2012

Number Nine

With a finite number of terms, it can only be approximated (to any conceivable degree of precision, mind you). I don't think there is any other algebraic expression.

11. Jun 6, 2012

Vorde

Well including the complex plane I'd reckon that $\frac{e^ix-cos(x)}{i}$ should be equal to sin(x), but I doubt thats what you are looking for. I don't think there is a non-infinite algebraic function that will accomplish what you are looking for.

12. Jun 6, 2012

mesa

Technically correct but not quite what I'm looking for, cos put's me back to square one lol

Interesting, well lets give it a shot.
Any thoughts on where to start? I was going to use the unit circle and partially filled areas to see what I can pull from that.

Last edited: Jun 6, 2012
13. Jun 6, 2012

HallsofIvy

Staff Emeritus
I still don't know what you mean by this. "y= sin(x)" is a perfectly good function and is every bit as "exact" as, say, $\sqrt{x}$ and $e^x$. So the question is what do you mean by "exact" here?

14. Jun 6, 2012

mesa

That it is, but I want to see what sin actually looks like algebraically, and it should be a fun exercise.

15. Jun 6, 2012

Number Nine

It doesn't look like anything, algebraically; it's a trigonometric function. The closest you can get is it's Taylor series, which has already been posted.

16. Jun 6, 2012

mesa

Most likely true. Worst case this is a good exercise for an older returning student. I want to get a strong handle on this stuff and I really enjoy digging into what is taught in my classes.

I have an idea about what I want to try and would love some input from people like you. Heck, you likely will have a completely different and better strategy than mine, but most importantly I will learn from this process!

Math is a big subject and I am regularly amazed at what it can accomplish at the hands of those who know how to wield it ;)

17. Jun 6, 2012

Vorde

Well I'd say the only luck you are going to get is with the taylor polynomial, but that won't tell you what sine 'actually looks like'- just how to mimic it to an arbitrary precision.

I can't see a way to write an equation for sine without using other trigonometric functions, but good luck to you!

18. Jun 15, 2012

mesa

Well I have to admit this one has been challenging, although I'm not throwing in the towel yet.

19. Jun 15, 2012

Number Nine

Well, you should. As has been clearly explained to you, sin has no expression in term of elementary algebraic operations. Trying to square the circle is not noble, just futile.

20. Jun 15, 2012

Bob S

Here is another way. Do you remember the Pythagorean theorem? A2 + B2 = C2, where C is the hypotenouse, and A is close to the origin?
Choose C=1, and using the angle between A and C, slowly increase the angle from 0 to 90 (or 180) degrees. The length of B is equal to the sine of the angle.