Lesson 30 – Trigonometric
Functions & Periodic
Phenomenon
Pre-Calculus
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Lesson Objectives
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1. Relate the features of sinusoidal curves to
modeling periodic phenomenon
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2. Transformations of sinusoidal functions
and their features
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(A) Key Terms
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Define the following key terms that relate to
trigonometric functions:
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(a) period
(b) amplitude
(c) axis of the curve (or equilibrium axis)
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(A) Key Terms
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The General Sinusoidal Equation
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In the equation f(x) = asin(k(x+c)) + d,
explain what:
a represents?
k represents?
c represents?
d represents?
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Contextual Analysis: DATA SET
•Follow this link for the graph of the data.
•Adjust the sliders to fit the data.
•What are each variable affecting?
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(C) Modeling Periodic Phenomenon &
Transformed Sinusoidal Curves
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(C) Modeling Periodic Phenomenon &
Transformed Sinusoidal Curves
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Examples to Develop
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Sketch  2 cycles
Analyze  (D, R, max/min, roots, period,
amplitude, axis of curve)
(a) the function
(b) the function
(c) the function
(d) thefunction

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
f (x)  sin( 2x)
g(x)  3cos(0.5x)
h(x)  0.5sin( 0.25x) 1

k(x)  2cos(3(x  ))  2
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(C) Modeling Periodic Phenomenon &
Transformed Sinusoidal Curves
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A spring bounces up and down according to the model
d(t) = 0.5cos(2t), where d is the displacement in centimetres
from the rest position and t is the time in seconds. The model
does not consider the effects of gravity.
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(a) Make a table for 0 ≤ t ≤ 9, using 0.5-s intervals.
(b) Draw the graph.
(c) Explain why the function models periodic behaviour.
(d) What is the relationship between the amplitude of the
function and the displacement of the spring from its rest
position?
(e) What is the period and what does it represent in the
context of this question?
(f) What is the amplitude and what does it represent in the
context of this question?
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(E) Combining Transformations
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We continue our investigation by graphing some other functions in
which we have combined our transformations
(i) Graph and analyze y  2sin 3(x  60 ) 1
 identify transformations and state how the key features have
changed
o

 
y  2cos2(x  ) 3
(ii) Graph and

4 
 analyze
 identify transformations and state how the key features have
changed
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1
 
y  tan x  
(iii) Graph and
2
4 
analyze
 identify transformations and state how the key features have
changed
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(B) Writing Sinusoidal Equations
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ex 1. Given the equation y = 2sin3(x - 60°) + 1,
determine the new amplitude, period, phase shift
and equation of the axis of the curve.
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Amplitude is obviously 2
Period is 2π/3 or 360°/3 = 120°
The equation of the equilibrium axis is y = 1
The phase shift is 60° to the right
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(B) Writing Sinusoidal Equations
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ex 2. Given a cosine curve with an amplitude of 2, a period of
180°, an equilibrium axis at y = -3 and a phase shift of 45°
right, write its equation.
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So the equation is y = 2 cos [2(x - 45°)] – 3
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Recall that the k value is determined by the equation period =
2π/k or k = 2π/period
If working in degrees, the equation is modified to period =
360°/k or k = 360°/period
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Writing sinusoidal equations
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Given a tangent curve with amplitude of 4,
period of 2π/3, reflected across the x-axis,
shifted up 5, and a phase shift of π/4
Given a sine curve with amplitude of ½,
period of 270°, phase shift of 60°
Given secant curve with amplitude of 3,
period of 6π, shifted down 2.
Given cosecant curve with amplitude of 4,
reflected across the x-axis, period of 4
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More Graphing with transformations
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More Graphing with transformations
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Write the equation from the graph
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Write the equation from the graph
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Write the equation from the graph
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Write the equation from the graph
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Write the equation from the graph
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Write the equation from the graph
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Applications of Sinusoidal Fcns
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Write the equation that can be used to model the following
relationship between a riders height (in m above the ground) and
time spent on the ride, in seconds since the rider started the ride.
NOTE: assume the rider got on the ride when the wheel was at its
lowest height.
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A Ferris wheel with a radius of 7 m, whose axle is 9 m above the
ground and that rotates once every 40 seconds.
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Use your equation to predict:
(a) the rider height after 15 seconds
(b) at what time(s) the rider is 14 m above the ground
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Applications of Sinusoidal Fcns
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The average monthly temperature, T(t), in degrees Celsius in Kingston,
Ontario, can be modeled by the function below, where t represents the
number of months (for t = 1, the month is January)

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T(t)  14.6sin x  4.2  5.9
6
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(a) Determine the period and state its significance in this context
(b) Determine
the equation of the axis of the curve and state its significance

in this context
(c) Determine the amplitude of the curve and state its significance in this
context
(d) Graph the function on your graphing calculator
(e) Evaluate T(30) and explain what your answer means in this context
(f) Solve T(t) = 12 and explain what your answer means in this context
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