# Workshop on Geometry Software

Math 241
Spring 2015
Professor: Mike O'Sullivan

### Week 4: Trigonometry

First Day: Trigonometric Functions and Geometric Constructions

I started with a review of similar triangles, and the proportionality of the sides. Then, I described how to find the point (x, 1/x) for arbitrary x using straightedge and compass. Well, we used higher level tools based on straightedge and compass, such as the construction of a perpendicular to a line through a given point.

I described how to find the point (t, tan(t)) for arbitrary t. We start with constructing an arbitrary arc of the unit circle, call that arc t. Then we use GeoGebra to make the line x=t, and from that point we use constructions based on straightedge and compass.

Assignment:
1. Make a worksheet in which you animate a point on the x-axis and use the construction of (x, 1/x) discussed in class to graph the function f(x)= 1/x.
2. Make a worksheet in which you animate a point on the unit circle and use the construction of (t, tan(t)) discussed in class to graph the function tan(x).
3. Make a worksheet in which you animate a point on the unit circle and graph the functions sin(x) and cos(x)

Second Day: Sums of sine Functions

We used the built in sine function today. We started by graphing f(x)= A*sin(w*x +b) using sliders for A, w, b. We explored the effect of varying each of the parameters: A, the amplitude, w the frequency, and b, which is related to the phase shift (the phase shift is -b/w).

Next we compared sin functions with different frequencies (different values of w) for example sin(x), sin(2*x), sin(3*x). We looked at the sum of the three functions and found the period of the sum. We repeated this for other combinations of sin functions, for example sin(x), sin(x/2), sin(x/3) and sin(x), sin(x/2), sin(x/4). We also explored the impact of using different amplitudes and values for b.

Assignment:
1. Prepare a worksheet in which you explore today's topic. I'm leaving this rather open so that you will try something interesting to you. What happens when you add 4 sine functions with different frequencies, amplitudes and phases? Can you connect properties of the sum of the functions with the summands? Make something pretty, use animation. Describe what you did so that someone else could reproduce it.