4.[N] Suppose S_{1}(t) is a sine wave of amplitude 1 and frequency 1 Hz,
and S_{2}(t) is a sine wave of amplitude 2 and frequency 2/3 Hz. Suppose
also that both are zero at t = 0 and start to increase as t begins to increase from 0.
(That is, S_{1}(t) = sin(6.28t) and S_{2}(t) = sin(4.19t).
How did the 6.28 and 4.19 get here? Remember sin(x) repeats after x increases from 0
to 2p (about 6.28) radians, so S_{1}(t) repeats after t increases from 0 to 1.)

(a) What are the values of S_{1}(t) at t = 0.75, 1.5, 2.25, 3, 3.75,
and 4.5 seconds? Answer

(b) What are the values of S_{2}(t) at t = 0.75, 1.5, 2.25, 3, 3.75,
and 4.5 seconds? Answer

(c) Denote by S_{3}(t) the sum S_{1}(t) + S_{2}(t).
What are the values of S_{3}(t) at t = 0.75, 1.5, 2.25, 3, 3.75,
and 4.5 seconds? Answer

(d) Guess the frequency of S_{3}(t). Answer

5. Suppose S_{1}(t) is a sine wave of amplitude 1 and frequency 1 Hz,
and S_{2}(t) is a sine wave of amplitude 2 and frequency 1/sqrt(2) Hz.
Suppose also that both are zero at t = 0 and start to increase as t begins to
increase from 0. Recall sqrt(2) is an irrational number, so we claim in
the text S_{3}(t) = S_{1}(t) + S_{2}(t) never repeats.
On the other hand, observe that S_{3}(0) = 0 and
S_{3}(0.390045...) = 0. What's going on here?
Answer

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