IDEAS ON TRIGONOMETRY

When you understand this animated diagram,
 You shall understand the sin(x):
Source: http://id.mind.net/~zona/mmts/trigonometryRealms/trigonometryRealms.html

We should help our teachers to TRULY UNDERSTAND what is going on here!
To my mind, it embodies not only the meaning of the sine function, but the possibilty of understanding the use of the trigonometric functions as models of real-life situations.
Given the emphasis on integration in the Curriculum 2005 philosophy, I believe we should help teachers understand the use of the trigonometric functions as models to study applications of real-life situations.
And in studying such applications, the important concepts in trigonometry are developed ...
Functions and modelling and applications should be the thread and unifying concepts in the mathematics curriculum ...

I found the following quote:
Once we get past the learning stage of finding components of vectors, the sine and cosine functions find great use in describing any kind of periodic or oscillatory motion, e.g. an object bobbing on the end of a spring, a vibrating guitar string or tuning fork, even a pendulum swinging back and forth. This kind of motion, harmonic motion, is found just about everywhere! Cycles govern almost every aspect of life - heartbeats, sleep patterns, seasons, and planetary orbits all follow regular, predictable cycles. The buzzwords that we use to describe this kind of movement are period, amplitude, frequency, and phase.
See:  http://dept.physics.upenn.edu/courses/gladney/mathphys/subsubsection1_1_4_2.html

Likewise:
You first met the trigonometric functions in algebra and trigonometry in high school. In a typical trigonometry course the functions ... are defined as ratios of sides in right triangles. The focus is on measuring the sides and angles of triangles, hence the term trigonometric functions. Much of your attention was directed to applications in geometry growing out of this connection with right triangles as well as to identities that express relationships between the several trigonometric functions.
In calculus the focus changes. The trigonometric functions are defined in terms of arclength on a unit circle, and the emphasis is on the periodic behavior of the trigonometric functions. It is their periodicity that leads to their most important applications in science -- modeling phenomena that repeat as a function of time. Simple harmonic motion (sinusoidal motion), light and sound waves, electricity, gravitational waves in the universe, oscillations of the pendulum of a clock, oscillations of atomic crystals on which our most accurate time keeping is based -- all these are periodic phenomena that are modeled mathematically by the trigonometric functions. The essential characteristics of any periodic motion are the amplitude (how big are the oscillations?) and period (how long is a single wave?). Sometimes the phase (how much has the wave been translated to the right or delayed in time?) is also important.
See:  http://www.dartmouth.edu/~matc/math3.calculus/CAS/CAS1.4/frame1.4.html

There are clearly two ideas here -- components in a right triangle, and periodic functions. Where should our emphasis be?

In the following, I throw several ideas on a heap to try to make the point that we should develop the idea of trigonometry as providing models for periodic phenomena ...

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