*pi*.

In class we talked about how to find *B* in the expression
*f*(*x*) = *A* cos(*B x*) and
*g*(*x*) = *A* sin(*B x*) so that the
functions *f*(*x*) and *g*(*x*) have a given
period. This web explanation tries to do that more carefully.

Let's consider the sine function. We've graphed sin(*x*) to the
right. It starts at (0,0), does a complete cycle, and finishes the
cycle at 2*pi* (because its period is 2*pi*). That is, when
we plug 2*pi* into the sine function, we get the second blue dot,
which is where the function starts repeating.

Now suppose we want the sine curve sin(*B* *x*) to have
a period of *k*. That means that we want the blue dot to appear
when *x* = *k*. That is, when we plug in
*x* = *k* we should get the blue dot:

sin(*B* *k*) = the blue dot.

How can this be? Well, the sine function gives us the blue dot when
the thing that was plugged in is equal to 2*pi* (as shown in the
figure). So what we plugged into our new sine function must be equal
to 2*pi*:

All we're doing here is figuring out how much we have to stretch or
shrink the graph horizontally to get it to end at the right place.
The way we do this is by making whatever we're plugging into the sine
(or cosine) equal 2*pi*.

horizontal stretching and trig functions

Last modified: Thu Sep 15 15:09:56 EDT 2005

Comments to:glarose(at)umich(dot)edu

©2005 Gavin LaRose, UM Math Dept.