Chapter 4 Trigonometric Functions

1
Chapter 4 Trigonometric Functions
4.5
Applications of Sinusoidal Functions
MATHPOWERTM 12, WESTERN EDITION
4.5.1
2
Sinusoidal Graphs
An important application of trigonometry is the
use of sinusoidal functions to model periodic
data. The horizontal axes of graphs of
trigonometric functions are marked with p and
its multiples. Graphs of trigonometric
functions may also have the horizontal axis
marked with numbers instead of multiples of p.
These graphs would be more useful for
applications involving sinusoidal patterns.
The horizontal axis uses numbers. The period of
the function has been adjusted to a period of 1.
The equation is y sin 2p t, where t represents
time.
4.5.2
3
Tracking the Height of the Tide
Level (m)
Amplitude 2
Vertical Displacement 3
Period
b
Time (h)
The maximum height of the tide is 5 m and the
minimum height is 1 m.
The period is 6 h.
The equation of this function is
4.5.3
4
Modelling Tides
The depth of water, d(t), in meters, in a
seaport can be approximated by the sine
function, d(t) 2.5sin 0.164p (t - 1.5) 13.4,
where t is the time in hours. Sketch the graph
of the function.
Depth of the water (m)
Time (h)
4.5.4
5
Modelling Tides contd
Maximum 15.9
Depth of the water (m)
12.2 h
8.751
0.346
Minimum 10.9
d(t) 2.5sin 0.164p (t - 1.5) 13.4
Time (h)
Amplitude
2.5 m
Phase Shift
1.5 units to the right
Period
Find the time for which the depth of the water is
at least 12 m
8.751 - 0.346 8.4 h for each cycle.
12.2 h
4.5.5
6
Sketching the Graph of a Ferris Wheel Ride
A Ferris wheel ride has a radius of 20 m and
travels at a rate of 6 revolutions per minute.
You board the ride at the bottom chair from a
platform 2 m above the ground.
a) Sketch a graph showing how your height above
the ground varies during the first two
cycles. b) Write an equation which expresses your
height as a function of the elapsed time. c)
Calculate your height above the ground after 12 s.
4.5.6
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Sketching the Graph of a Ferris Wheel Ride
contd
Height (m)
Time (s)
The equation is
The height after 12 s is 15.9 m.
or
4.5.7
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Another Ferris Wheel Ride
This graph shows the height, h, in metres above
the ground, over time, t, in seconds, for a
Ferris wheel ride.
a) What is the period for 1 revolution of the
ride? b) How high is the centre of the Ferris
wheel off the ground? c) Write the equation
using the sine function for h in terms of t. d)
Find the persons height on the ride at t 10 s.
e) Find the first time, in seconds, that a
person is 6 m above the ground.
4.5.8
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Another Ferris Wheel Ride contd
a) The period is 32 s.
b) The centre of the Ferris wheel is 10 m off the
ground.
c) The equation, using the sine function for h in
terms of t, is
d) The height of the person when t 10 is 13.4
m because
e) Using the graph and tracing, the person is 6 m
above the ground after 5.63 s.
4.5.9
10
Assignment
Suggested Questions
Pages 225-227 1-5 odd, 6-15, 18, 20, 22
4.5.8