CIRCULAR MOTION AND SIMPLE HARMONIC MOTION (SHM)

 As an essential background you should first thoroughly study this online page:
Note: In all sketches, left-click the red circle and drag ...!
A day at the track

Then also read and understand:
Simple harmonic motion

Also browse this webpage - the concepts are sound and it has good applications:
http://people.hofstra.edu/faculty/Stefan_Waner/trig/trigintro.html

Parametric equations
 We are NOT working with circular motion, but with its components!
Of course we must understand the point P(x, y) on the circle with radius A is given by P(Acos q, Asin q). The ordered pairs (Acos q, Asin q) describe a circle, and going twice around the circle is good to define q + 360 = q.

But what we must understand is that cos q describes the horizontal distance of P from (0,0) as q changes.
So cos (q ) is a function in q ....
Likewise sin q describes the vertical distance of P from (0,0.)

The following sketch illustrates that cos q is not so much connected to circular motion, but to oscillatory motion!

The horisontal distance covered by the point moving on the circle is the projection on to the x-axis. Now look at this projection:


Our interpretation of this simple harmonic motion gives the possibility of many connections and new understandings. For example, for me, personally, the signs of sin q and cos q is "suddenly" intuitively as clear as daylight:
Ball A above starts at 1 and moves to zero, so this distance is positive (so cos q is positive in 1st quadrant), then it moves from 0 to -1, so this distance is negative (so cos q is negative in 2nd quadrant), then it moves from -1 to 0 (so cos q is negative in 3rd quadrant), then it moves from 0 to 1 (so cos q is positive in 4th quadrant). Likewise, for the vertical oscilation, the ball moves from 0 to 1, from 1 to 0, from 0 to -1, from -1 to 0, defining the signs of sin q in the four quadrants.

But periodic or oscillatory motion are usually rather associated with time as independent variable and not the angle. We can easily make this transition: If we understand the "angle" as an arc length, i.e. measured in radians, the distance the ball travels on the arc is given by wt, where w is its angular speed (this speed w is given by 2p/T, where 2p is the distance covered in one cycle and T is the period, the time for one cycle). If the ball starts with an initial distance of d on the arc (the initial phase), we can express the horisontal distance as x(t) = Acos(wt + d).

It is this model that gives rise to our general function y = acos(bq+c), i.e. these phenomena are the reason for studying such functions!

If we have have graphing calculators available, we can experience this circular motion! The TI-83 screenshot below shows the graph of the two parametric functions
X = 3cos(T)
Y = 3sin(T)

We can easily check that in
X = 3cos(2T)
Y = 3sin(2T)
the point travels twice as fast, i.e. it covers twice the distance in the same time - the connection between the speed w and the period of the graph is an eye-opener! Likewise, understanding the beginning phase can give concrete meaning to graphs such as y = cos(q + 30)!

Lets now experience this circular motion in the following Java applets.
The first example below is a circle defined by x(t) = rcos(t) and y(t) = rsin(t).
We have the added advantage that we can trace the curve ...
Note that the "time" interval runs from 0 to 2pi, i.e. one revolution and the "speed" (the number of intervals) is the same for all examples.
Click on the drop-down menu at the top and Trace to illustrate how the graph is influenced by
- amplitude (or vertical scaling), e.g. 5cos(t),
- period (or horisontal scaling), e.g. 10cos(5t),
- phase (or horisontal translation), e.g. 10cos(t+1), and
- vertical translation, e.g. 10cos(t)+5
The challenge is to see the connections between the diagram, its defining formulae, and the separate graphs of x(t) and y(t),

Activity: For each example in the drop-down menu, click Trace Curve!, study the movement and then describe and graph x(t) and y(t).
Remember, x(t) is the projection of the movement of the point onto the X-axis, and y(t) is the projection of the path of the point onto the Y-axis.
Note: you can experiment further by entering your own formulae and changing the values of t (defining the domain and range!).

Challenge
The examples below are very interesting curves! Click Trace Curve! and watch the movement!
The movements may seem complicated, but, still, x(t) merely describes the projection of the movement of the point onto the X-axis and y(t) the movement onto the Y-axis, and we should be able to describe, graph and give the generating formula of this projection! For example, in the Connected hearts curve below, despite the complicated point movement, x(t) simply is a cosine function with amplitude 10, repeating 3 times in the time interval 0 to 2pi, so its formula is x(t) = 10cos(3t).
For each of the functions in the drop-down menu below, click on Trace Curve!, analyse the movement of the point carefully, then draw the graphs of x(t) and y(t) and give their defining formulae:

Need help or want to check your answers?
The launcher below contains the same examples, but also shows the generating formulae. So you can deduce the behavior of x(t) and y(t) by graphing their formulae - if needed, you can draw the graphs of x(t) and y(t) using a Simple JavaGraph applet or the Function Grapher below.
I "invented" these curves by simply varying the formulae. You can experiment by entering your own formulae ...

SOME ACTIVITIES:

SHM:. The PENDULUM and and a SPRING PENDULUM are examples of SIMPLE HARMONIC MOTION, where the distance, speed, ... vs time is modelled by functions of the form y = acos(bt+c). Take a close look at these very interesting interactive illustrations:
Click here for the pendulum applet
Click here for the spring applet

Tides:
Click here for an (unedited) activity on tides Shaheeda included in Unit 3.

Seasonal temperature:
Click here for an Excel activity on seasonal temperature I designed.

Music:
Virtual reality Java piano: the relationship between the sounds and the underlying mathematics and physics
Trigonometry and music: frequency, amplitude, ...

Kinematics: An important problem in kinematics is to understand the motion of a point on the rim of a wheel.
Suppose the wheel has radius a, and is rolling from left to right along the x-axis at a speed of a units/second, thus, the angular velocity of the points on the rim of the wheel is 1 radian/second with respect to the center of the wheel. Then a point on the rim of the wheel is going around a circle, as the center of the circle is itself moving along the line y = a. Deduce that the parametric equations of the path of a point on the rim of the wheel are
x = at + acos(t)
y = a + asin(t)
Note: We see here again that we cannot work in degrees - the quantity "at" has no meaning if "t" is in degrees!

IMPLICATIONS
 All this, to my mind, says we should start with CO-ORDINATE definitions of the trigonometric functions, not with the triangle, which is a special case ..... I am not motivating it much, except to say that it is very difficult to generalise to the general case from the triangle. See this effort to try do it in a sound way:
http://www.ucl.ac.uk/Mathematics/geomath/trignb/trigmod.html

Also, we should do radians, because you cannot really do applications in terms of time if you do not have radians as the "angle" unit! This, to my mind, is an example of how changes in teachnology should influence the curriculum: the availability of these new technology tools means that it is now possible to do these applications - but the applications and the technology itself requires the use of radians. Therefore the curriculum should change to include radians ...

When, of course, we do do right triangle trigonometry, I strongly suggest we do it through problems, like here:
http://www.mste.uiuc.edu/tcd/Trig/frame.html
Note: You need Geometer's Sketchpad to work through the activities, but it is great!

This is also a good introduction:
http://aleph0.clarku.edu/~djoyce/java/trig/

INTERACTIVE TOOLS FOR INVESTIGATION
There are many online applets showing interactive trigonometry concepts, you should really visit these sites and reflect on how it can be used:
http://www.ies.co.jp/math/java/trig/
http://www.bun.falkenberg.se/gymnasium/amnen/matte/trigapplets/trigfunc.html
http://mathinsite.bmth.ac.uk/html/applets.html

Here is an important onsite applet - it carries important meanings!
Sin-Cos-Tan Definitions vs graphs

Here are three Excel activities I have quickly designed, trying to capture the same ideas:
Definitions
y=asinb(q - c) + d
y=acosb(q - c) + d

Function Grapher: Here is an online function grapher, that we can use to explore functions and check our conjectures. For example, enter 3cos(2x) and click Graph it -- note, it is case sensitive! Why not do it now?

Sorry, but you need a Java-enhanced browser to use this function grapher.

Click here for help on how to use the applet.

We can illustrate the functions with Geometer's Sketchpad (you need Geometer's Sketchpad to open it):
Trigonometric functions

Implications
The point is that the availability of these graphing tools radically changes the possibilities of teaching and learning. The power at our fingertips means that we can give teachers the opportunity to approach any idea in an investigative, exploratory way. This then simultaneously reinforces their content knowledge and embodies the very processes of mathematical activity!

Names and notations
I think we should help teachers to realise that NAMES and NOTATIONS do not come first.
In this case, like in all of mathematics, we study the relationship between variables (functions) and many times we are interested in things that are constant despite variation (e.g. direct proportion, the sum of the angles of a triangle, Pythatgoras, etc.). So here we have:
that something like y/x is constant despite the variance in y and x ...
This turns out to be a function of q . We can/should in the beginning denote it, like other functions, as f(q ).
This is an interesting function, because although we do not have a "formula", and mathematicians for centuries did not have a formula, we can nevertheless find f(30) = 0,5, f(31) = 0,513, etc.

This function is so special that mathematicians gave it a special name, so we denote it sin(q ).
Only few functions are so special that they have special names. Another one is log(x).
I am pushing that we write sin(q ) and log(x) for a long time before changing to sin q and log x!!!! I am not going to motivate it much, except to say that that is the historical development, and maybe it will help people understanding trigonometry as FUNCTIONS and stop such howlers as sin(A+B) = sin A + sin B !!