Trying to figure out a single variable expression for Sin

[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?

 

[mathman]

You need to clarify what you have in mind.

single variable expression for Sin

 

[Number Nine]

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?

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?

 

[HallsofIvy]

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

 

[mesa]

You need to clarify what you have in mind.

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

 

[mesa]

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?

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?

 

[Number Nine]

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?

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.

 

[HallsofIvy]

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

 

[mesa]

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.

Is it possible to get an exact function for sin?

 

[Number Nine]

Is it possible to get an exact function for sin?

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.

 

[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 that's what you are looking for. I don't think there is a non-infinite algebraic function that will accomplish what you are looking for.

 

[mesa]

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

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

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.

Interesting, well let's 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.

 

[HallsofIvy]

Is it possible to get an exact function for sin?

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?

 

[mesa]

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?

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

 

[Number Nine]

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

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.

 

[mesa]

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.

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 ;)

 

[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!

 

[mesa]

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

 

[Number Nine]

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

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.

 

[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.

 

[mesa]

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.

This is supposed to be a learning exercise ;)

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.

I've played with pythagorea's but realized that would just give me a function of y for a given x when the Sine curve is based on a value for y based on the length of the radian created on a unit circle. Could play around with deformations and such with pythagorea's but I thought I would try something else first.

The formula for calculating the area of a partially filled circle is based on a value for θ while the formula for the area of a partially filled sphere is based on a value for our y. If I can convert the first formula into 3 dimensions while preserving θ I could try to pull out what we need from the two by setting them equal to each other.

Here is what I have so far:

∏ x r^2 - ((∏ x r^2/2) + (∏ x r^2(2(θ/ 2 x Pi)) + (sin θ)(cos θ))

It calculates the area of a partially filled circle (at least one that is less than half full lol) with θ being the angle created by the dividing line of the circle to the radius to the top of the segment for the area which we are calculating for. It spits out the right number for area given the restriction above.

Next I wanted to see what took a circle to a sphere so divided (4/3)∏r^3 by ∏r^2 resulting in 4/3r. Unfortunately I can't just take 4/3r and multiply it into the formula above; that would be like taking the area of the circle segment, setting it equal to a smaller circles' area and then calculating for the volume of a sphere of that smaller circles' dimension of radius. Basically useless in most cases.

I noticed that ∏ can be looked at as a constant (2Pi really but that's besides the point). I was thinking I could use a function of 4/3r and the arc length to get my third dimension and get our θ based formula.

Any thoughts, and/or corrections?

Trig is remarkable! :)

 

[Millennial]

Finding a polynomial expression for sine is not really possible. All of equivalent sine definitions take sine as a transcendental function that does not have an expansion in terms of polynomials or algebraic expressions. I will still give some definitions for sine, maybe you might be interested in one of them.

$\displaystyle \sin(x)=\sum_{k=0}^{\infty}(-1)^k\frac{x^{2k+1}}{(2k+1)!}$
$\displaystyle \sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$ where i is the imaginary unit.
$\displaystyle \sin(x)=\Re(e^{ix})$
$\displaystyle \sin(x)=\frac{\pi}{\Gamma(x/\pi)\Gamma(1-x/\pi)}$ where $\Gamma(x)$ denotes the Gamma function.

Sine is the only solution to the differential equation $\displaystyle \frac{d^2 y}{d x^2}+y=0$ satisfying $y(0)=0,\,y'(0)=1$.

Sine is the only solution to the integral equation $\displaystyle \int_{0}^{f(a)}\frac{dx}{1-x^2}=a$ for the function f.

You should be able to derive more by yourself.