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Trigonometric functions

It will become obvious that the trigonometric functions, sine, cosine, and tangent in particular, are the most frequently used functions you will see in introductory physics. The most important trigonometric identities are listed in your supplementary information sheet. Although you will have plenty of practice in resolving vectors into components using these functions, we will list here, as a reminder, the basic definitions then move on to more sophisticated uses of the trignometric functions for describing oscillations. Be sure to brush up on some basic trigonometry and using right-angled triangles.

For mathematics and physics, we rarely use degrees as arguments of the trigonometric functions but radians, where, as usual the definition of the radian, as shown in the figure above, is

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. To see an example of what they mean, consider the plot of the sine function below. You can adjust the amplitude, frequency, and phase to see how they change the look of the plot.

We will see these later in the course, but since repetition breeds familiarity, we will define them now as long as we're talking about useful functions and then tie these buzzwords to physical examples later. The best way to get up close and familiar with the definitions is to see them in action. Just so you start to think about 3-dimensions instead of the ordinary 2, here's an example of what Maple can do quite handily,

Parametric plot of [x,y,z] = [cos(x)-2*cos(0.4*y), sin(x)-2*sin(0.4*y), y]

Hence, on to Maple...

#############################################################################
#                  Maple and Trigonometric Functions
#
#Let's start by looking at the basic sine function plotted from 0 to
#2*Pi radians.
#
plot(sin(theta), theta=0..2*Pi);
#
#Now we define the frequency and amplitude operationally - i.e. we show
#what they do by putting them in and changing them.  The amplitude is
#represented by the letter A and the frequency by f.
#
y := A*sin(f*theta);
#
#Note that if the amplitude A is 1 and the frequency f is 1, then we have
#the same function we first plotted.  Let's use Maple's subs command to
#make several plots with varied values of f with A fixed at 1.
plot(subs(A=1, f=1, y), theta=0..2*Pi);
plot(subs(A=1, f=2, y), theta=0..2*Pi);
plot(subs(A=1, f=3, y), theta=0..2*Pi);
plot(subs(A=1, f=.5, y), theta=0..2*Pi);
#
##############################################################################
#We can animate all these as follows:
with(plots):
animate(subs(A=1, y), theta=0..2*Pi, f=0.5..3);
##############################################################################
#
#Now vary the amplitude, holding f fixed at 1.
plot(subs(A=2, f=1, y), theta=0..2*Pi);
plot(subs(A=3, f=1, y), theta=0..2*Pi);
plot(subs(A=.5, f=1, y), theta=0..2*Pi);
#
##############################################################################
#For animation fans - you can animate the amplitude in the same
#                     way as we did the frequency.
#
animate(subs(f=1, y), theta=0..2*Pi, A=0.5..3);
##############################################################################
#
#Hence we see that the amplitude represents the maximum value (both positive
#and negative) for the function.  The frequency determines the number of
#oscillations for a range of radian values.  The period relates to the
#amount of time it takes for the function to go through one full cycle.
#In the present case, we see that sine returns to its origin condition when
#theta becomes 2*Pi.  The number of oscillations in that range is 1.  The
#period is defined to be 1/f.  The phase is trickier.  To see it, we note
#that, in some instances, we don't want the value of the function at the
#origin to be zero.  So we have to add something to the argument of sine
#to make it so.  Here's the operational example.
#
y := A*sin(f*theta + phase);
plot(subs(A=1, f=1, phase=0, y), theta=0..2*Pi);
plot(subs(A=1, f=1, phase=Pi/2, y), theta=0..2*Pi);
plot(subs(A=1, f=1, phase=Pi, y), theta=0..2*Pi);
plot(subs(A=1, f=1, phase=3*Pi/2, y), theta=0..2*Pi);
plot(subs(A=1, f=1, phase=2*Pi, y), theta=0..2*Pi);
#
##############################################################################
#The animation for the phase should be easy for you by now
##############################################################################

To regain some familiarity with the trigonometric functions, do exercises 1-28 and 1-29. Do exercises 1-30 and 1-31 to see a way of making an infinite number of new functions using the standard set we have introduced.

Next: Parametric Equations Up: The Most Useful Functions Previous: Exponentials