graphing trig functions
TRANSCRIPT
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Unit 7: Trigonometric Functions
Graphing the Trigonometric Function
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CCSS: F.IF. 2, 4, 5 &7E; F.TF. 1,2,5 &8
E.Q: E.Q1. What is a radian and how do I us it tod t r!in an"# ! asur on a $ir$# %
2. ow do I us tri"ono! tri$ 'un$tions to!od # ( riodi$ ) ha*ior%
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Mathematical Practices:
1. Make sense of problems and persevere in solving them.2. Reason abstractly and quantitatively.3. Construct viable arguments and critique the reasoning ofothers.
4. Model ith mathematics.!. "se appropriate tools strategically.#. $ttend to precision.%. &ook for and make use of structure.'. &ook for and e(press regularity in repeated reasoning.
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Right Triangle Trigonometry
SOHCAH
TOA
CHOSHACAO
Graphing the Trig Function
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5
Graphing Trigonometric Functions
Amplitude : the maximum or minimum verticaldistance bet een the graph and the x!axis" Amplitude is al ays positive
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The amplitude of y = a sin x (or y = a cos x) is half the distance between the maximum and minimum values of the function.
amplitude = |a | If |a | > 1, the amplitude stretches the raph verticall!.If " # | a | > 1, the amplitude shrin$s the raph verticall!.If a # ", the raph is reflected in the x%axis.
&
'
&
(
y
x
(
&
y = sin xreflection of y = sin x y = sin x
y = sin x
&1 y = sin x
y = & sin x
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Period : the number o# degrees or radians e mustgraph be#ore it begins again"
Graphing TrigonometricGraphing Trigonometric
FunctionsFunctions
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y
x &
sin x y = period* & &sin = y
period*
The period of a function is the x interval needed for thefunction to complete one c!cle.
+or b > ", the period of y = a sin bx is .b
&
+or b > ", the period of y = a cos bx is also .b
&
If " # b # 1, the raph of the function is stretched hori ontall!.
If b > 1, the raph of the function is shrun$ hori ontall!. y
x & ' ( cos x y =
period* &&1
cos x y = period*
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The sine #unction
sin
0 0
/2 1
0
3/2 1
2 0
y
x
II
I I I IV
I
45
90
135
180
270
225
0
315
90 180 2700 360
I II
III IV
sin
magine a partic!e on the unit circ!e" #tarting at $1"0% an& rotatingcounterc!oc'(i#e aroun& the origin) *+er, po#ition o- the partic!ecorre#pon (ith an ang!e" ." (here , #in .) # the partic!e mo+e#through the -our ua&rant#" (e get -our piece# o- the #in graph ) From 0 to 90 the , coor&inate increa#e# -rom 0 to 1 ) From 90 to 180 the , coor&inate &ecrea#e# -rom 1 to 0 ) From 180 to 270 the , coor&inate &ecrea#e# -rom 0 to 1
) From 270 to 360 the , coor&inate increa#e# -rom 1 to 0
nteracti+e ine n(rap
http://www.dynamicgeometry.com/JavaSketchpad/Gallery/Trigonometry_&_Analytic_Geometry/Sine_Wave_Geometry.htmlhttp://www.dynamicgeometry.com/JavaSketchpad/Gallery/Trigonometry_&_Analytic_Geometry/Sine_Wave_Geometry.html -
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$ine is a periodic #unction: p % &'
,Domain
One period
2
0 322 3
sin
sin : omain $ang!e mea#ure#% a!! rea! num er#" $ " % :ange $ratio o- #i&e#% 1 to 1" inc!u#i+e ;1" 1 co# .) # the partic!e mo+e#through the -our ua&rant#" (e get -our piece# o- the co# graph ) From 0 to 90 the > coor&inate &ecrea#e# -rom 1 to 0 ) From 90 to 180 the > coor&inate &ecrea#e# -rom 0 to 1 ) From 180 to 270 the > coor&inate increa#e# -rom 1 to 0
) From 270 to 360 the > coor&inate increa#e# -rom 0 to 1
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-raph of the osine +unction
To s$etch the raph of y = cos x first locate the $e! points.
These are the maximum points, the minimum points, and theintercepts.
10101co# x
0 x &
&'
&
Then, connect the points on the raph with a smooth curvethat extends in both directions be!ond the five points. /sin le c!cle is called a period .
y
&'
& &&
' &
&0
1
1
x
y = cos x
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(osine is a periodic #unction: p % &'
One period 2
32 23 0
cos
cos : omain $ang!e mea#ure#% a!! rea! num er#" $ " % :ange $ratio o- #i&e#% 1 to 1" inc!u#i+e ;1" 1i#)
cos() = cos(),Domain
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Properties o# $ine and (osine graphs
)" The domain is the set o# real numbers&" The rage is set o# *y+ values such that !), y ,)
-" The maximum value is ) and the minimum valueis !)
." The graph is a smooth curve/" 0ach #unction cycles through all the values o# the
range over an x interval or &'1" The cycle repeats itsel# identically in both
direction o# the x!axis
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$ine Graph
Gi en ! A sin " x Amp#itu$e % I AIperio$ % &'( "
)xamp#e!y%5 sin &*
Amp% 5
+erio$% &'(& % '
''(&'( -'(
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)xamp#e!y%&cos 1(& *
Amp% &
+erio$% &'( .1(&/ '
(osine Graph
Gi en ! A sin " xAmp#itu$e % I AIperio$ % &'( "
'&'' -'
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y
1
1&'
&
x ' & (
)(ample * $etch the raph of y = ' cos x on the interval 2 , 3.4artition the interval 2", & 3 into four e5ual parts. +ind the five $e!
points6 raph one c!cle6 then repeat the c!cle over the interval.
ma> x intmin x intma>
30303y 3 co# x 20 x &
&
'
(", ')
&' ( , ")
( , ")&
&( , ')
( , ')
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y
x &
y = cos ( x)
7se basic tri onometric identities to raph y = f ( x))(ample * $etch the raph of y = sin ( x).
7se the identit!sin ( x) = sin x
The raph of y = sin ( x) is the raph of y = sin x reflected in the x%axis.
)(ample * $etch the raph of y = cos ( x).
7se the identit! cos ( x) = cos x
The raph of y = cos ( x) is identical to the raph of y = cos x.
y
x &
y = sin x
y = sin ( x)
y = cos ( x)
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&
y
&
8
x&
8
0
'
'
&
8
8
'
&
'
&
020 20y 2 #in 3 x
0 x
)(ample * $etch the raph of y = & sin ( ' x).
9ewrite the function in the form y = a sin bx with b > "
amplitude * |a | = | &| = &
alculate the five $e! points.
(", ") ( , ")'
( , &)&
( , %&)8
( , ")'
&
7se the identit! sin ( x) = sin x: y = & sin ( ' x) = & sin ' x period *
b
& &'=
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Tangent Function
sin cos tan
/2
/4
0
/4
/2
tan
/2 un&
/4 1
0 0
/4 1
/2 un&
:eca!! that )
ince cos i# in the &enominator" (hen cos 0" tan i# un&e-ine&)Thi# occur# ? inter+a!#" o--#et , /2 @ A /2" /2" 3/2" 5/2" A B
CetD# create an >/, ta !e -rom = /2 to = /2 $one inter+a!%"
(ith 5 input ang!e +a!ue#)
cossin
tan =
&&
&&
&&
&&
1
1
1
1
1
0
0 0
un&
un&
0
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Graph o# Tangent Function: Periodic
tan
/2n& $ %
/4 1
0 0
/4 1
/2 n&$ %
,Domain tan i# an odd -unction= it i# #,mmetric (rt the origin)
tan() = tan()
0
tan
/2 /2
Ene perio&
tan : omain $ang!e mea#ure#% . /2 n :ange $ratio o- #i&e#% a!! rea! num er# $ " %
3/23/2
ertica! a#,mptote# (here co# . 0
cossintan =
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y
x
&
'
&'
&
&
-raph of the Tan ent +unction
&. 9an e* ( , : )'. 4eriod*
. ;ertical as!mptotes*( ) += k k x
&
1.
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&. +ind consecutive vertical
as!mptotes b! solvin for x*
. $etch one branch and repeat.
)(ample * +ind the period and as!mptotes and s$etch the raphof x y &tan
'1=
&&,
&&
== x x
,
== x x;ertical as!mptotes*
)&
,"( '. 4lot several points in
1. 4eriod of y = tan x is .
&
. is&tanof 4eriod x y =
x
x y &tan'
1==
'
1 "
"=
'
1=
'
'
1
y
x&
=
'
(
= x
(
= x
'1
,
'
1,=
'1,'
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(otangent Function
sin cos cot
0
/4
/2
3/4
cot
0 n&
/4 1
/2 0
3/4 1
n&
:eca!! that )
ince sin i# in the &enominator" (hen sin 0" cot i# un&e-ine&)
Thi# occur# ? inter+a!#" #tarting at 0 @ A " 0" " 2" A B
CetD# create an >/, ta !e -rom = 0 to = $one inter+a!%"
(ith 5 input ang!e +a!ue#)
#inco#
cot =
&&
22
&&
2
2
0
1
0
0
1
1
1 0
n&
n&
H1
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Graph o# (otangent Function: Periodic
cot
0
/4 1
/2 0
3/4 1
,Domain cot i# an odd -unction= it i# #,mmetric (rt the origin)
tan() = tan()
cot : omain $ang!e mea#ure#% . n :ange $ratio o- #i&e#% a!! rea! num er# $ " %
3/23/2
ertica! a#,mptote# (here #in . 0
#inco#
cot =
/2 /2
cot .
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-raph of the otan ent +unction
&. 9an e* ( , : )'. 4eriod*
. ;ertical as!mptotes*( ) = k k x
1.
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(osecant is the reciprocal o# sine
sin : omain $ " % :ange ;1" 1pposite Angle
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$impli#y each expression"
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Using the identities you no 9no 6 #indthe trig value"
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sinD % !)B-6 ) o H D H &7 o I #ind tanD
secD % !7B/6 ' H D H -'B&I #ind sinD
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Si!i#ariti s and /i'' r n$ s
a5 o do you #ind theamplitude and period #orsine and cosine #unctionsC
b5 o do you #ind theamplitude6 period andasymptotes #or tangentC
c5 2hat process do you#ollo to graph any o# thetrigonometric #unctionsC