Trigonometry
Sinusoidal Modeling Problems
Suppose that you are on the beach at Port Aransas, Texas. At 2:00 p.m. on March 19, the
tide is in (i.e., the water is at its deepest). At that time you find that the depth of the water
at the end of the pier is 1.5 meters. At 8:00 p.m. the same day when the tide is out, you
find that the depth of the water is 1.1 meters. Assume that the depth varies sinusoidally
with time.
a) Derive an equation expressing the depth of the water in terms of the number of
hours that have elapsed since 12:00 noon on March 19th.
b) Use your mathematical model to predict the depth of water at
i) 4:00 p.m. on March 19,
ii) 7:00 am on March 20,
iii) 5:00 p.m. on March 20.
c) At what time will the first low tide occur on March 20?
d) Approximate the earliest time on March 20 that the water will be 1.27 meters
deep.
Circadian rhythm is the daily biological pattern by which body temperature, blood
pressure, and other physiological variables change. The data show typical changes in
human body temperature over a 24-hour period ( t = 0 corresponds to midnight). Find a
cosine curve that models the data.
When two species interact in a predator /prey relationship, the population of both species
tend to vary in a sinusoidal fashion. In a certain midwestern county, the main food source
for barn owls consists of field mice and other small mammals. The table below gives the
population of barn owls in this county every July 1 over a 12-year period. Find a sine
curve that models the data.
Buried treasure Problem. You seek a treasure that is buried in the side of a mountain.
The mountain range has a sinusoidal cross-section. The valley to the left is filled with
water to a depth of 50 meters and the top of the range is 150 meters above the water level.
You set up an x-axis at the water level and a y-axis 200 meters to the right of the deepest
part of the water. The top of the mountain is at x = 400 meters.
(a) Write an equation y in terms of x for points on the surface of the
mountain.
(b) Show by calculation that the origin satisfies this equation.
(c) The treasure is located within the mountain at the point (x, y) = (130,40).
Which would be a shorter way to dig to the treasure,
a horizontal tunnel or a vertical tunnel? Justify your answer.
Spacecraft Orbiting: Problem: When a spaceship is fired into orbit from a site such as
Cape Canaveral, which is not on the equator, it goes into an orbit that takes it alternately
north and south of the equator. Its distance from the equator can be approximated by a
sinusoidal function of time. Suppose that a spaceship is fired into orbit from Cape
Canaveral. Ten minutes after it leaves the Cape, it reaches its farthest distance north of
the equator, 4000 kilometers. Half a cycle later it reaches its farthest distance south of the
equator, also 4000 kilometers. The spaceship completes an orbit (and therefore one cycle)
every 90 minutes. Letting y represent the number of kilometers the spaceship is north of
the equator and t the number of minutes that have elapsed since takeoff, write an
equation expressing y in terms of t . Sketch a graph of y versus t, labeling all axes
carefully. Predict the distance of the spaceship from the equator when
t = 25 , t = 41, and t = 163 . Calculate the distance of Cape Canaveral from the equator.