APPLICATIONS OF SINUSOIDAL PHENOMENON
1. An economist indicated that demand for temporary employment (measured in thousands of job
applications per week) in your county can be modeled by the function y  4.3sin 0.82x  0.3  7.3
where x is time in years since January, 1995. Calculate the amplitude, the vertical offset, the phase
shift, and the period, and interpret the results.
Sinusoidal models are of great importance to meteorological studies, as when modeling the temperature
based on the time of day, the illumination of the Moon as it goes through its phases, or even the prediction of
tidal motion. Below are some examples.
Tidal Waves: Tsunamis, also known as tidal waves, are ocean waves produced by earthquakes or other
upheavals in the Earth’s crust and can move through the water undetected for hundreds of miles at
great speed. While traveling in the open ocean, these waves can be represented by a sine graph with a
very long wavelength (period) and a very small amplitude. Tsunami waves only attain a monstrous size
as they approach the shore, and represent a very different phenomenon than the ocean swells created
by heavy winds over an extended period of time.
2. A graph modeling a tsunami wave is given below. What is the height of the tsunami wave (from crest
to trough)? Note that h=0 is considered the level of a calm ocean. What is the tsunami’s wavelength?
Find the equation for this wave.
3. A heavy wind is kicking up ocean swells approximately 10 feet high (from crest to trough), with
wavelengths of 250 feet. Find the equation that models these swells. Determine the height of a wave
measured 200 feet from the trough of the previous wave?
4. The graph below shows the deviation from the average daily temperature for the hours of a given day,
with t=0 corresponding to 6 A.M. Use the graph to determine the related equation. Use the equation
to find the deviation at 5P.M. If the average temperature for this day was 72◦, what was the
temperature at midnight?
 
5. The equation y  7sin  t  models the height of the tide along a certain coastal area, as compared to
6 
average sea level. Assuming t=0 is midnight, predict the height of the tide at 5 A.M. Is the tide rising
or falling at this time?
Many animals exhibit a wavelike motion in their movements, as in the tail of a shark as it swims in a straight
line or the wingtips of a large bird in flight. Such movements can be modeled by a sine or cosine function and
will vary depending on the animal’s size, speed, and other factors. Below are some examples.
6. The graph below models the position of a shark’s tail at time t, as measured to the left (negative) and
right (positive) of a straight line along its length. Use the graph to determine the related equation. Is
the tail to the right, left, or at center when t=6.5 sec? How far? Would you say the shark is “swimming
leisurely” or “chasing its prey”? Justify your answer.
7. The State Fish of Hawaii is the humuhumunukunukuapua’a (reef triggerfish), a small colorful fish found
abundantly in coastal waters. Suppose the tail motion of an adult fish is modeled by the equation
d t   sin15 t  with d(t) representing the position of the fish’s tail at time t (in seconds), as measured
in inches to the left (negative) or right (positive) of a straight line along its length. Is the tail to the left
or right of center at t=2.7 seconds? How far? Would you say this fish is “swimming leisurely” or
“running for cover”? Justify your answer.
8. The typical voltage V supplied by an electrical outlet in the U.S. is a sinusoidal function that oscillates
between -165 volts and +165 volts with a frequency of 60 cycles per second. Obtain an equation for
the voltage as a function of time t.
9.
Mark Twain sat on the deck of a river steamboat. As the paddle wheel turned, a point
on the paddle blade moved so that its distance y, in feet from the water’s surface was a
sinusoidal function of time t, in seconds. When Twain’s stopwatch read 4 seconds, the
point was at its highest, 16 ft above the water’s surface. The wheel’s diameter was 18
ft, and it completed a revolution every 10 seconds.
What is the lowest the point goes? Why is it reasonable for this value to be negative?
Find an equation for distance (in feet from the water’s surface) as a function of time. (HINT – Make a sketch)
How far above the surface was the point when Mark’s stopwatch read 17 seconds?
What is the first positive value of t at which the point was at the water’s surface? At that time was the point
going into or coming out of the water? How can you tell?
10. Zoey is at summer camp. One day she is swinging on a rope tied to a tree branch, going back and forth
alternately over the land and water. Nathan starts a stopwatch. At time x=2 seconds, Zoey is at one end of her
swing at a distance y = -23 feet from the river bank (see the figure). At time x=5 seconds, she is at the other end
of her swing at a distance y=17 feet from the riverbank. Assume that while she is swinging y varies sinusoidally
with x.
Write an equation to describe the distance from the riverbank. (Hint sketch graph first).
When x= 13 seconds, was Zoey over land or over water? Explain how you know.