Sine Cosine: Model Waves
'Model the ride of the Sine and Cosine Waves!'
The whole of the modern world is founded on our ability to be able to define perfectly the position, and movement, of waves. Mobile phones, X-ray machines, TVs, Telecommunications etc. harness the waves of light, sound, x-ray, gamma rays etc. in the physical world around us.
To gain a first insight into how we define waves, see if you can adjust the coefficients, a,b,c,d in the function: y=asin[b(x-c)]+d and y=acos[b(x-c)]+d to model the movement of a snake, the roof of a building, the formation of a wave and . . . can you work out what these 'undulations' are?
As you start changing a coefficient ask yourself: “What stays the same?”, “What is changing?”. The below video doesn't have any sound, but gives you a quick overview of the activities:
Resources
Use your knowledge of transformations of graphs to model these waves. Try a cosine function for the first two and a sine function for the last two (or experiment and decide for yourself whether sine or cosine would be best, and why.
The below links take you to larger versions of the above applets, but depend on your security settings as to whether or not they will be visible in your browser:Sine Function: 
Undulations Wave Formation
Cosine Function: Air France Slithering Snake
Description
- It's a good idea if students have completed the activity
Trig Transformations before attempting this modelling activity. - Open any one of the above four Geogebra applets in the "Resources" section above.
- Move the sliders to change the values of 'a', 'b', 'c' and 'd', one at a time, see if you can work out what effect each one has on the graph.
- What stays the same? What remains constant?
- When do you think a Cosine function is more appropriate than a Sine function? Does it matter?
- For extension, or as an alternative to some of the files above, students may like to find their own images to model:
Flickr Trigonometric Waves - A good activity to try following this one is Which Wave or Modelling Music.