Chapter 10 Trigonometric functions - Ugrad.math.ubc.ca

Math 102 Notes Chapter 10
In the expression above, the number 13 represents a shift along the time axis, and carries units of
time. We can express this same function in the form
( πt
H(t) = 50 + 50 sin
12 − 13π )
.
12
In this version, the quantity
φ = 13π
12
is what we have referred to as a phase shift. (This represents the point on the 2π cycle at which
the function begins when we plug in t = 0.)
In selecting the periodic function to use for this example, we could have made other choices.
For example, the same periodic can be represented by any of the functions listed below:
( π
)
H(t) = 50 − 50 sin
12 (t − 1) ,
( π
)
H(t) = 50 + 50 cos
12 (t − 19) ,
( π
)
H(t) = 50 − 50 cos
12 (t − 7) .
All these functions have the same values, the same amplitudes, and the same periods.
Example 3: phases of the moon
0 29.5
Figure 10.8: Periodic moon phases
A cycle of waxing and waning moon takes 29.5 days approximately. Construct a periodic function
to describe the changing phases, starting with a “new moon” (totally dark) and ending one cycle
later.
Solution:
The period of the cycle is T = 29.5 days, so
ω = 2π
T = 2π
29.5 .
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