Download Math Analysis AB

Math Analysis AB
Using Sine and Cosine Functions as Models
Worksheet 2.2
Warm Up
Find the amplitude, period, and phase shift for the sine function whose graph is shown. Write an
equation for this graph.
Now try finding a cosine function with this same graph.
MATHEMATICAL MODELING
Sine and cosine functions can be used to model many real-life situations including electric
currents, musical tones, radio waves, tides, sunrises, and weather patterns.
Example: Throughout the day, the depth of water at the end of a dock varies with the tides. The
table shows the depth (in meters) at various times during the morning.
t (time)
Midnight 2 a.m.
4 a.m.
6 a.m.
8 a.m.
10 a.m.
Noon
y (depth) 2.55
3.80
4.40
3.80
2.55
1.80
2.27
a. Use a trigonometric function to model this data.
b. Find the depth at 9 a.m. and 3 p.m.
c. A boat needs at least 3 meters of water to moor at the dock. During what times in the
afternoon can it safely dock?
Math Analysis AB
Modeling Problems
1. Household electrical power in the US is provided in the form of alternating current.
Typically the voltage cycles smoothly between +155.6 and -155.6 volts 60 times per
second. Use a cosine function to model the alternating voltage.
2. A rabbit population in a national park rises and falls each year. It is at its minimum of
5000 rabbits in January. By July, as the weather warms up and food grows more
abundant, the population triples in size. By the following January, the population again
falls to 5000 rabbits, completing the annual cycle. Us a trigonometric function to find a
possible formula for R = f(t) where R is the size of the rabbit population as a function of
t, the number of months since January.