Chapter 6 Periodic Functions

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Chapter 6Periodic Functions 6.1 The Sine and
Cosine Functions 6.2 Circular Functions and
their Graphs 6.3 Sinusoidal Models 6.4
Inverse Circular (Trigonometric Functions)
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• Periodic Functions
• There is a whole class of natural behaviors whose
  most striking characteristic is their cyclical
  nature.
• tides
• phases of the moon
• daily temperature
• sleep-wake cycles
• insect invasion
• seasonal sales

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6.1 The Sine and Cosine Functions
KEY The unit circle with equation p2 q2 1

• Two ways to specify point on the circle
• p,q coordinates
• distance around circle

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Using p, q coordinates
Suppose that a point P with coordinates (0.5, q)
is on the unit circle. Find the values of q.
Suppose that a point P with coordinates (p, 0) is
on the unit circle. Find the values of p.
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• Using distance around circle
• starting point (1,0)
• wrap counter-clockwise

Suppose that a point P is located 2p units around
circle. Find the coordinates of P.
Suppose that a point P is located p units around
circle. Find the coordinates of P.
Suppose that a point P is located -p/2 units
around circle. Find the coordinates of P.
p ( 3.14) is a really NICE number when it comes
to measuring arcs
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• Using distance around circle
• starting point (1,0)
• wrap counter-clockwise

directedarc -2p -3p/2 -p -p/2 0 p/2 p 3p/2 2p
terminalpoint
Let t represent the length of the arc traversed
counter-clockwise from (1,0) Find terminal point
corresponding to t 3p and t -7p/2 and t
2004p. Find terminal point corresponding to t 3
and t -5 and t 100.
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Defining the Sine and Cosine Functions
The sine function, sin(t), is the second
coordinate (output), q, of the unit-circle point
(p,q) associated with the arc t (input).
The cosine function, cos(t), is the first
coordinate (output), p, of the unit-circle point
(p,q) associated with the arc t (input).
sine functionwrap t and identify (p,q)
t
q
cosine functionwrap t and identify (p,q)
p
t
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Evaluating sine and cosine
tarc -2p -3p/2 -p -p/2 0 p/2 p 3p/2 2p
terminal point
sin(t)
cos(t)
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6.2 Circular Functions and their GraphsBehavior
of Sine Function
As t increases from 0 to p/2, sin(t) ____________
from ___ to ___ (rapidly at first but then
slowly).
As t increases from p/2 to p, sin(t) ____________
from ___ to ___ (slowly at first but then
rapidly).
As t increases from p to 3p/2, sin(t)
____________ from ___ to ___ (rapidly at first
but then slowly).
As t increases from 3p/2 to 2p, sin(t)
____________ from ___ to ___ (slowly at first
but then rapidly).
As t increases from 2p to 5p/2, sin(t)
______________ from ___ to ___
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Graph of f(t) sin(t)
local max? local min?
domain? range?
intercepts?
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6.2 Circular Functions and their GraphsBehavior
of Cosine Function
As t increases from 0 to p/2, cos(t) ____________
from ___ to ___ (slowly at first but then
rapidly).
As t increases from p/2 to p, cos(t) ____________
from ___ to ___ (rapidly at first but then
slowly).
As t increases from p to 3p/2, cos(t)
____________ from ___ to ___ (slowly at first
but then rapidly).
As t increases from 3p/2 to 2p, sin(t)
____________ from ___ to ___ (rapidly at first
but then slowly).
As t increases from 2p to 5p/2, sin(t)
______________ from ___ to ___
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Graph of f(t) cos(t)
domain? range?
intercepts?
local max? local min?
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Comparison of Sine and Cosine Graphs
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The connection between sine and cosine
We know sin(t) is the q coordinate. We know
cos(t) is the p coordinate. We know p2 q2 1.
Trig Identitysin(t)2 cos(t)2 1 sin2(t)
cos2(t) 1
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