Thread: Sine Waves

Okay,

I am studying maths at a university (home study)

I cant get my head around sine waves. Right, this is what I know:

A sine-wave is called a sine wave because if we stick a triangle in a circle and make it go round and round and compare anan angle of a triangle in the circle to the height of one of the sides as it goes round and round and plot all of these differences on a graph, we will get a bouncy up and down wave which we call a "sine wave".

Okay great, I dont see any useful application for this - if you plot radio frequency, or a clean sound frequency on an oscilloscope you will also get a "sine wave", but what that got to do with triangles and circles?

and why the circle and triangle in the first place? :S

Sorry if my description gets a little messy. It's kind of hard to write in a way that would help you visualize something, but I tried lol. Hope it helps:


There is no real relation between the triangles and actual waves. The triangle simply helps us calculate points that would lie on the circle.
All sine is, is an oscillating function. If you graph it you see it going up and down, but that is with (usually the x axis, lets call it time) progressing, so it may be tempting to look at the shape of this graph, but if you ignore the graph and just think about what sine does as you plug in different values, all it does is go up and down between 1 and -1. So in itself sine is a very useful function it can describe many things. In fact you don't even need cosine, you can just use sine but add (1/2pi) to the angle and you would get the same thing. So:
sin(theta) + cos(theta) = sin(theta) + sin(theta+(1/2)pi)

Now imagine the circle graphed with sine and cosine. Imagine that you are lying on the plane of the page looking down at the circle, it would just appear as a straight line, then follow what happens to the circle as "the triangle is traced around" all you would see is a single point oscillating back and forth. Then if you looked at it from a side view you would just be seeing the effect of sine and again would see a point oscillating up and down. So they are two independent oscillating functions that make the circle, and it works because they are a half step off.

If we take the two functions to be vectors, then and you add the two vector values you will be putting one oscillating line at the end of the other, and the end point will trace out a circle.


So to answer your question a circle is kind of just a graph we can make with sine AND cosine. By themselves they are just functions that moves back and forth which is why we can apply sine or cosine to many things in physics.

Originally Posted by leohopkins

Okay,

I am studying maths at a university (home study)

I cant get my head around sine waves. Right, this is what I know:

A sine-wave is called a sine wave because if we stick a triangle in a circle and make it go round and round and compare anan angle of a triangle in the circle to the height of one of the sides as it goes round and round and plot all of these differences on a graph, we will get a bouncy up and down wave which we call a "sine wave".

Okay great, I dont see any useful application for this - if you plot radio frequency, or a clean sound frequency on an oscilloscope you will also get a "sine wave", but what that got to do with triangles and circles?

and why the circle and triangle in the first place? :S

Forget the word "wave" for the time being. Go look at a graph of the sine function. You can find such a graph in any calculus or trigonometry book.

You can describe the sine function in terms of circles and angles. Take a circle of radius 1 centered at the origin (0,0) in the plane and consider a line segment from the origin to the circle. Consider the angle between the x-axis and that line segment, call that angle theta. Then the cosine of theta is the x-coordinate of the intersenction of the line segment with the circle and the sine is the y-coordinate.

Here is fairly good expositi0on of trigonometric functions, with some useful pictures. http://en.wikipedia.org/wiki/Trigonometric_function

Okay thanks guys, i still dont fully understand it, but i shall study it until i do

The reason math is developed in one way or another is that someone wanted to solve a problem. In the past the only people paying for the work were the royalty.

So now you ask about sine waves. Well, a sine wave is a plot. It's a visual. Like they say a picture is worth a 1000 words.

Sine is really cool. It's a ratio. It relates the length of the far side of a right triangle to the length of the hypotenuse. So this ratio is related to the angle. Now think about this. Sometimes that ratio is negative. Lengths are always positive. So sometimes we measure the length of the far side of the triangle as a minus number.

So here is how that wave comes about. You draw a graph of the angle and the sine. You get a 'wave'. It's a neat looking curve, but it's just a plot. If the circle has radius 1 and I use distance along the circle you mentioned as the angle, then I get some really nice properties. Angles begin at 0. At pi over 2 the plot is at its highest. At pi the plot is back to 0. This form of angular measurement is called radians. The math gets simpler. That is why its used. Math guys ahve enough problems. Anything that makes things easier is welcome.

Anyways, hope this is a start.

Originally Posted by hokie

The reason math is developed in one way or another is that someone wanted to solve a problem. In the past the only people paying for the work were the royalty.

So now you ask about sine waves. Well, a sine wave is a plot. It's a visual. Like they say a picture is worth a 1000 words.

Sine is really cool. It's a ratio. It relates the length of the far side of a right triangle to the length of the hypotenuse. So this ratio is related to the angle. Now think about this. Sometimes that ratio is negative. Lengths are always positive. So sometimes we measure the length of the far side of the triangle as a minus number.

So here is how that wave comes about. You draw a graph of the angle and the sine. You get a 'wave'. It's a neat looking curve, but it's just a plot. If the circle has radius 1 and I use distance along the circle you mentioned as the angle, then I get some really nice properties. Angles begin at 0. At pi over 2 the plot is at its highest. At pi the plot is back to 0. This form of angular measurement is called radians. The math gets simpler. That is why its used. Math guys ahve enough problems. Anything that makes things easier is welcome.

Anyways, hope this is a start.

You probably ought to take a look at the web site posted a bit earlier, or better yet, go read a book on trigonometry. You have oversimplified the issue a bit, and distorted the notion of sine in the process.

If you really want to approach the sine from a fundamental perspective it can be done by taking as your definition , where is defined by the usual power series. If you would like to see this done, you might take a look at Walter Rudin's Real and Complex Analysis.

Originally Posted by leohopkins

Okay great, I dont see any useful application for this -

Understanding sine waves is enormously useful. You cannot begin to understand alternating current, electrical generation, three-phase power, impedance, transmission lines, electronic filters, or anything of that nature without understanding sine waves. In the mechanical engineering world, it is the basis for analyzing anything with springs and masses, structures and their response to earthquakes, wind, etc.

Originally Posted by Harold14370

Originally Posted by leohopkins

Okay great, I dont see any useful application for this -

Understanding sine waves is enormously useful. You cannot begin to understand alternating current, electrical generation, three-phase power, impedance, transmission lines, electronic filters, or anything of that nature without understanding sine waves. In the mechanical engineering world, it is the basis for analyzing anything with springs and masses, structures and their response to earthquakes, wind, etc.

Not to mention the very useful tricks involving Fourier series and Fourier transforms to study the behavior of dynamical systems under fairly arbitrary inputs in terms of the response to sines and cosines.

Originally Posted by leohopkins

Okay great, I dont see any useful application for this - if you plot radio frequency, or a clean sound frequency on an oscilloscope you will also get a "sine wave", but what that got to do with triangles and circles?

Perhaps you might want to concentrate in areas in which trigonometric functions are not of great importance.

How would you feel about a career as a shepherd?

Just to throw my 2 cents in, I'd say the three most commonly used areas of math are arithmetic, algebra and trigonometry, in that order. Arithmetic is used every time you add two numbers, count out a group of objects, or really just about anything else. Algebra is used any time you try to fit data into a formula (plugging in $21 and 15% to get the tip) or trying to alter a formula to fit the data (changing a celsius-to-fahrenheit forumla to fahrenheit-to-celcius). Trigonometry is used in just about everything that involves either triangles, circles or periodic events (just think of how much that covers).

You probably ought to take a look at the web site posted a bit earlier, or better yet, go read a book on trigonometry. You have oversimplified the issue a bit, and distorted the notion of sine in the process.

Dr Rocket I probably know as much about math as you do. The thread here is to assist a beginner. Thanks for the references, but I read my first book on trig 35+ years ago.

I can assure you that the cis function that you have referenced is something I knew about many decades ago, but is hardly the starting point for anyone beginning to get a feel for math.

Having read a bit further along the thread I saw:

How would you feel about a career as a shepherd?

A shabby comment. Is this due to an inferiority complex?

Originally Posted by leohopkins

Okay,

I am studying maths at a university (home study)

I cant get my head around sine waves. Right, this is what I know:

A sine-wave is called a sine wave because if we stick a triangle in a circle and make it go round and round and compare anan angle of a triangle in the circle to the height of one of the sides as it goes round and round and plot all of these differences on a graph, we will get a bouncy up and down wave which we call a "sine wave".

Okay great, I dont see any useful application for this - if you plot radio frequency, or a clean sound frequency on an oscilloscope you will also get a "sine wave", but what that got to do with triangles and circles?

and why the circle and triangle in the first place? :S

Sorry I took so long, I do a lot of other stuff. This is from the Bureau of Naval Personnel's Navy training course printed 1960. It is the training course for the Navy. On Sine Waves.

An alternator basically puts out this wave form.











Sincerely,


William McCormick

Originally Posted by hokie

Dr Rocket I probably know as much about math as you do. The thread here is to assist a beginner. Thanks for the references, but I read my first book on trig 35+ years ago.

Maybe. Maybe not.

Wht makes you think so?

Originally Posted by hokie

A shabby comment. Is this due to an inferiority complex?

No.

Originally Posted by hokie

I can assure you that the cis function that you have referenced is something I knew about many decades ago, but is hardly the starting point for anyone beginning to get a feel for math.

CIS function -- sounds like you must be a software guy.

That function is better known as the Euler formula. It is quite a bit older that 35 years, as one might guess from the name. It is quite possibly the single most important function in mathematics and physics. Electrical engineers use it on a daily basis. From the representation of sines and cosines in terms of the complex exponential you can derive quite a bit. It is, for instance, quite a good starting point for the development of the theory of one complex variable.

(Most interesting math thread in ages for (me). I'd wondered the same things.)

What is the cis function?

Originally Posted by Arcane_Mathamatition

Precisely.

A most beautiful and useful function.

One can develop a tremendous amount of mathematics and physics starting from this expression. In The Feynman Lectures on Physics Feynman gives a virtuoso performance using Euler's formiula.

It is also a useful starting point in the development of a good deal of complex analysis. It allows a thorough axiomatic treatment of trigonometry, easilty traceable to the foundations of mathematics in the natural numbers. It is also a very nice way to show the fundamental trigonometric identies from the easily derivable formulas:

and

How do you derive the formula?

Originally Posted by thyristor

How do you derive the formula?

The easiest way is to start with the usual power series expansion for the exponential function



From whence



And then breaking this up into real and imaginary parts,



Then either by definition or from elementary calculus you recognize the power series expansion for sine and cosine to conclude that

Originally Posted by Harold14370

Originally Posted by leohopkins

Okay great, I dont see any useful application for this -

Understanding sine waves is enormously useful. You cannot begin to understand alternating current, electrical generation, three-phase power, impedance, transmission lines, electronic filters, or anything of that nature without understanding sine waves. In the mechanical engineering world, it is the basis for analyzing anything with springs and masses, structures and their response to earthquakes, wind, etc.

Okay, well. What I DO understand of a sine wave plot/graph is, the closer together the peaks and troughs the higher the frequency and therefore the more energy there is. Likewise, the higher the peaks and troughs, the more amplitude there is and therefore there is more energy present there too.

But when presented with say a frequency of a graph of either sound or light (for example) - How are the sine, cosine, tangent etc functions useful?. What I mean is, its a graph; it has an x/y axis, you look at the value of those. The graphs plots go up and down uniformally, so you can tell that the data is cyclic in nature; but you look at time and you look at the y-axis and you extract from that any information you need; without the need for the sine funciton. - Perhaps im missing something?

Originally Posted by leohopkins

Originally Posted by Harold14370

Originally Posted by leohopkins

Okay great, I dont see any useful application for this -

Understanding sine waves is enormously useful. You cannot begin to understand alternating current, electrical generation, three-phase power, impedance, transmission lines, electronic filters, or anything of that nature without understanding sine waves. In the mechanical engineering world, it is the basis for analyzing anything with springs and masses, structures and their response to earthquakes, wind, etc.

Okay, well. What I DO understand of a sine wave plot/graph is, the closer together the peaks and troughs the higher the frequency and therefore the more energy there is. Likewise, the higher the peaks and troughs, the more amplitude there is and therefore there is more energy present there too.

But when presented with say a frequency of a graph of either sound or light (for example) - How are the sine, cosine, tangent etc functions useful?. What I mean is, its a graph; it has an x/y axis, you look at the value of those. The graphs plots go up and down uniformally, so you can tell that the data is cyclic in nature; but you look at time and you look at the y-axis and you extract from that any information you need; without the need for the sine funciton. - Perhaps im missing something?

I think you are missing quite a lot.

First if you look at the signal as being something like current or voltage then the power is represented by the square of the signal, and the integral of a sine or cosine squared, over the interval from 0 to is one, quite independent of the frequency. That fact is crucial to recognizing that sines and cosines form a complete set of orthonormal functions over that interval which in turn is the critical observatin necessary to the development of the theory of Fourier series.

All of the trig functions turn out to be useful in many branches of engineering and physics. Just resolving vector forces in specified directions requires the trig functions -- take a look at any engineering text on statics or dynamics. Trig functions and Euler's formula are also critical to the theory or ordinary AC circuits -- talk to any electrical engineer. Similarly resolution of light spectra by frequency is critical to optics. Quantum mechanics, in its more elementary forms, is known as "wave mechanics" and again the trigonometric functions are critical. Surveyors live and die with calculation that are totally dependent on trigonometry. The theory of differential equations, particularly second-order ordinary differential equations with constant coefficients is intimately tied with sines and cosines -- this is because sines and cosines are derivatives of one another which in turn is related to the Euler formula discussed earlier (exponential functions are their own derivativesa). Modern control theory is equally concerned with eigenfunctions of differential operators and again trig functions are important.

The theory of Fourier series allows one to study relatively general functions by representing them as a sum of sines and cosines. Then by knowing how a (linear) system reacts to sines and cosines one can determine how it will react to a more general function. This is an extremely powerful technique which generalizes to the theory of the Fourier transform, which again is critical in modern communication theory and information theory. Again, talk to any elelctrical engineer.

It is much harder to find a discipline in which trig functions are not important than to find one in whic they are critical. Shepherds don't need trigonometry very much.

Originally Posted by leohopkins

Originally Posted by Harold14370

Originally Posted by leohopkins

Okay great, I dont see any useful application for this -

Understanding sine waves is enormously useful. You cannot begin to understand alternating current, electrical generation, three-phase power, impedance, transmission lines, electronic filters, or anything of that nature without understanding sine waves. In the mechanical engineering world, it is the basis for analyzing anything with springs and masses, structures and their response to earthquakes, wind, etc.

Okay, well. What I DO understand of a sine wave plot/graph is, the closer together the peaks and troughs the higher the frequency and therefore the more energy there is. Likewise, the higher the peaks and troughs, the more amplitude there is and therefore there is more energy present there too.

But when presented with say a frequency of a graph of either sound or light (for example) - How are the sine, cosine, tangent etc functions useful?. What I mean is, its a graph; it has an x/y axis, you look at the value of those. The graphs plots go up and down uniformally, so you can tell that the data is cyclic in nature; but you look at time and you look at the y-axis and you extract from that any information you need; without the need for the sine funciton. - Perhaps im missing something?

You will have to show me how higher frequency at similar peak voltage is going to get you more RMS voltage, or amperage. Or even average voltage.


I have noted effects however, I believe capacitors in the circuit or capacitors ability to better supply and recharge in a certain voltage range, causes a slight difference. I also noted that a transformer created for a certain hertz seems to make less use of power at a higher hertz. It cannot make use of the supply, so it appears to draw less in some cases.


Sincerely,


William McCormick

(Great thread)