Mathematical Investigations IV
Name:
Mathematical Investigations IV
Trigonometry - Beyond the Right Triangles
And the Beat Goes On…
The Beat Phenomenon
1. Now it’s time to see how our work may be applied to analyzing the beat phenomenon. Let’s
begin by deriving the relevant formula.
a. First, add
sin(   )  sin(   ) 
b. You should have noticed some cancellation. To take advantage of this, solve the system
    x,     y ,
for  and  .
c. Now substitute back into your expression in (a), giving
sin x  sin y 
2.
Now let’s see an application of this formula. Let’s consider two pure sounds of
frequencies 20 Hz and 16 Hz. Below, write the corresponding sine functions which would
represent those pitches. Use t (for time) as the variable.
20 Hz:
16 Hz:
Trig. 8.1
Rev. S10
Mathematical Investigations IV
Name:
3. Next, add these two sine functions using the formula you just derived.
4. The red curve below is a graph of the sum of the two sine functions you found in Problem 2.
Notice that the formula suggests that this sum may be rewritten as a product of a sine function
and a cosine function. We’ll explore this in the next few questions.
2
1
0.2
0.4
0.6
0.8
1.0
1
2
a.
Write an equation for the sine function represented by the blue graph below. (Note:
There are only two graphs here. Blue and red make purple when the graphs are close together.)
2
1
0.2
0.4
0.6
0.8
1.0
1
2
Trig. 8.2
Rev. S10
Mathematical Investigations IV
Name:
b.
Write an equation for the cosine function represented by the green graph.
2
1
0.2
0.4
0.6
0.8
1.0
1
2
c.
Multiply your answers to (a) and (b) and compare with (3) above. What does this
indicate about the relationship between these three graphs?
2
1
0.2
0.4
0.6
0.8
1.0
1
2
Trig. 8.3
Rev. S10
Mathematical Investigations IV
Name:
5. Now that we’ve looked at the mathematics of the sum of the sines, let’s take a moment to
think about the physics.
a.
By comparing the red with the blue curve, what can you say about the frequency (in Hz)
of the resulting sound?
b.
The green curve suggests that the amplitude of the sound is changing; getting softer, then
louder, etc. What is the frequency (in Hz) of this sound change?
c.
The number of beats we hear each second is actually double the frequency you found in
(b). Briefly explain why.
d.
How is the number of beats you hear related to the frequencies of the two original
sounds?
e.
Given our discussion of sound and musical tones so far, do you think you would be able
to hear this sound?
Trig. 8.4
Rev. S10
Mathematical Investigations IV
Name:
6. Now let’s see if you can apply what you’ve learned. Using the same method as in (1), find a
formula for cos x  cos y.
7. Suppose you heard a sound at 422 Hz and heard six beats per second. What would the
frequency of the sounds be that would create this phenomenon? Check your work in
Mathematica.
8. Suppose you played sounds at frequencies of 400.6 Hz and 401 Hz. How many beats would
you hear in 10 seconds? First, work it out. Then check with Mathematica.
9.You play two sounds at a Hz and b Hz. You hear three beats every four seconds.
a. Find the relationship between a and b which would produce this phenomenon.
b. Using 400 Hz for a, find b. Check with Mathematica that you do in fact hear three beats
every four seconds.
Trig. 8.5
Rev. S10