APPLICATIONS OF SINUSOIDS
1. Will Rogers spun a lasso in a vertical circle. The diameter of the loop was 6 feet, and
the loop spun 50 times each minute. If the lowest point on the rope was 6 inches
above the ground, write an equation to describe the height of this point above the
ground after t seconds.
2. A weight in a spring-mass systems exhibits harmonic motion that can be described by
a sinusoid. The system is in equilibrium when the weight is motionless. If the weight
is pulled down or pushed up and released it would tend to oscillate freely if there were
no friction. In a certain spring-mass system, the weight is 5 feet from a 10 foot
ceiling when it is at rest. The motion of the weight can be described by the equation


y  3sin   t   , where y is the distance from the equilibrium point, and t is measured

2
in seconds.
a. Find the period of this motion.
b. Find the amplitude of this motion.
c. Was the weight pulled down or pushed up before being released?
d. How far was the weight from the ceiling when it was released?
e. How far from the ceiling is the weight after 2.5 seconds?
f. How close will the weight come to the ceiling?
g. When does the weight first pass its equilibrium point?
h. What is the greatest distance the weight will be from the ceiling?
3. The number of hours of daylight in one year in Ellenville can be modeled by a
sinusoidal function. During 2006, the longest day occurred on June 21 with 15.7
hours of daylight. The shortest day of the year occurred on December 21 with 8.3
hours of daylight. Write a sinusoidal equation to model the hours of daylight in
Ellenville.
4. In predator-prey situations, the number of animals in each category tends to vary
periodically. A certain region has pumas as predators and deer as prey. The number
of pumas varies with time according to the function P  200sin(0.4t  0.8)  500 and the
number of deer varies according to the function D  400sin  0.4t   1500 with time t, in
years.
a. How many pumas and deer will there be in the region in 15 years?
b. What is the maximum population of the pumas? When does this occur? What is
the minimum population of the pumas? When does this occur?
c. What is the maximum population of the deer? When does this occur? What is the
minimum population of the deer? When does this occur?
d. Using graphs of these two equations, make a statement regarding the relationship
between the number of puma and the number of deer.
5. A satellite is deployed from a space shuttle into an orbit which goes alternately north
and south of the equator. Its distance from the equator over time can be approximated
by a sine wave. It reaches 4500km, its farthest point north of the equator, 15 minutes
after the launch. Half an orbit later, it is 4500km south of the equator, its farthest
point south. One complete orbit takes 2 hours.
a. Find an equation of a sinusoidal function that models the distance of the satellite
from the equator.
b. How far away from the equator is the satellite 1 hour after launch?
6. Tarzan is swinging back and forth on his grapevine. As he swings, he goes back and
forth across the riverbank, going alternately over land and water. Jane decies to
model his movement mathematically and starts her stopwatch. Let t be the number of
seconds the stopwatch reads and let y be the number of meters Tarzan is from the
riverbank. Assume that y varies sinusoidally with t and that y is positive when Tarzan
is over water and negative when he is over land.
Jane finds that when t=2 Tarzan is at the end of his swing, where y=-23. She finds
that when t=5 he reaches the other end of the swing and y=17.
a. Sketch a graph of this function.
b. Write an equation expressing Tarzan’s distance from the river bank in terms of t.
c. Find y when t=2.8, t=6.3, and t=1.5.
d. Where was Tarzan when Jane started her stopwatch?
7. A pet store clerk noticed that the population in the gerbil habitat varied sinusoidally
with respect to time in days. He carefully collected data and graphed his resulting
equation. From the graph, determine the amplitude, period, phase shift and vertical
shift of the graph. Write the equation of the graph.
8. The temperature in an office is controlled by an electronic thermostat. The

11 
temperatures vary according to the sinusoidal function: y  6sin  x 
  19
 12
12 
where y is the temperature (◦C) and x is the time in hours past midnight.
a. What is the temperature in the office at 9AM when the employees come to work?
b. What are the maximum and minimum temperatures in the office?
9. A team of biologists have discovered a new species in the rain forest. They note the
temperature of the animal appears to vary sinusoidally over time. A maximum
temperature of 125◦ occurs 15 minutes after they start their examination. A minimum
temperature of 99◦ occurs 28 minutes later. The team would like to find a way to
predict the animal’s temperature over time in minutes. Your task is to help them by
creating a graph of one full period and an equation of temperature as a function over
time in minutes.
10. The angle of inclination of the sun changes throughout the year. This changing angle
affects the heating and cooling of buildings. The overhang of the roof of a house is
designed to shade the windows for cooling in the summer and allow the sun’s rays to
enter the house for heating in the winter. The sun’s angle of inclination at noon in
central New York state can be modeled by the formula
3600 
 360
Angle of Inclination (in degrees)  23.5cos 
x
  47 , where x is the number
365 
 365
of days elapsed in the year, with January 1 represented by x=1, January second
represented by x=2, and so on.
a. Find the sun’s angle of inclination on Valentine’s Day.
b. On what date of the year is the angle of inclination at noon the greatest?