Understand Sinusoidal Waves: Problem Solving

[johndb]

Problem understanding waves. I can see how rates of speed in circular motion can translate to expressed different kinds/shapes of waves but I don' see why concepts of trigonometry like sin,cos are brought into describing waves. I just feel there could be clearer ways of describing waves.

 

[symbolipoint]

Waves are characterized with periodicity, therefore Trigonometry functions are necessary to describe their features (period, amplitutide, phase shift, wavelength, frequency).

 

[johndb]

Trigonometry is necessary to describe triangles an amateur mathematician like myself doen't immediately see why they're used to describe circles and curves. Unless it is meant to describe a rotated circle, curve so you're viewing a limited, reduced amount of the circle, so could be said to be viewing the circle at an angle? Follow me?

 

[sethz]

The sin and cos of an angle are just the coordinates of a point rotated by that angle around a unit circle (sin is y, cos is x). The radian measure of an angle is just the distance traveled by that point around the circumference of the circle. So circles are basic to trigonometry, and the periodic nature of rotating around and around a circle is useful for describing the periodic nature of waves. (See the pic in the attachment.)

 

[johndb]

Elaborate on how sin, sohcahtoa, e.g. opposites divided by hypotenuse correspond and relate/interact to aspects of the waves . That still isn't enough, you say sin is x and cos is y...eh...Okay and I know what a radian is. So do I picture an arc as the opposite in sin(of sohcahtoa) and the hypotenuse will be the radius length. And a cos wave takes the adjacent and hypotenuse parts. Do these interact with regards to component speeds that correspond to the differently shaped waves? This is never so clear, I've never seen the connection made or the parts broken down in such a way, where am I going wrong?

 

[russ_watters]

The image sethz posted is graphed in polar coordinates and regular x/y coordinates, showing how you convert one to the other. A sine wave, on the other hand, is graphed with the angle on the x axis.

 

[LURCH]

The key thing is that voltages are all relative, a particular charge is called "zero," and the sinwave travels above and below that. So all the important information about the wave can be boiled down to different ways of asking two questions; "How far above/below zero is the wave at this point," and "How much farther until the wave crosses zero again?" Nearly evrything one could want to know about a sinwave can be answered with that information.

Now, these two questions are basically asking for the location of a point along the wave in two directions; vertical and horizontal. As these two directions are 90^o from one another, any attempt to compare them to each other will result in a right triangle (or "trigon,"). So the natural choice for discribing these relationships is trigonometry.

 

[johndb]

Where is the traditional trigonometric s.o.h process of taking e.g. an opposite value and divided it by the hypotenuse. Is this information in statements like e.g. the wave sin 2x. Where the 2 is divided by 1 say for s.o.h. I've seen and read examples that include angular velocity and the resulting wave, animations online e.g. wikipedia all very impressive but the integration of trigonometry still escapes me. I'm sure this seems very basic to many but I am stuck on this lesson. This probably tests the limits of how successful lessons are communicated online but I haven't access to such in person expertise at the moment. I appreciate any more insights people can offer thank you.

 

[rcgldr]

Perhaps this animation from Wiki showing the relationship between a circle and sine wave might help:

http://en.wikipedia.org/wiki/Sine_wave#See_also

 

[symbolipoint]

johndb,
Explaining the relationship between waves and Trigonometry is not easy to do for you through a couple of forum messages. If you successfully pass a course of Intermediate Algebra, then you are ready to study a course on Trigonometry. In this course, you will learn about angles, triangles, especially right-triangles, and circular functions; you will learn to relate angles and triangles to a unit circle. You will study graphs of the Trigonometric functions. While you study graphs of the functions, you will understand much of what you have been asking. On the other hand, you can also trying checking some of the webpage articles, in case they give you some satifying understanding sooner. You will probably be able to understand better by actually studying Trigonometry (college course or from textbook on your own).

 

[sethz]

To see the right-angle triangle, look at the unit circle below in which a radius has been rotated counterclockwise by theta from an initial position pointing to the right. The blue line is the "opposite" side of the triangle, the red line is the "adjacent" side, and the black line (radius) is the "hypotenuse". Note that since this is a unit circle, the hypotenuse (radius) is one. Therefore, opp/hyp is simply opp, which is just the y-component of the point P.

http://img99.imageshack.us/img99/1015/trig2mu1.gif