Graphing Trig Functions

7/26/2019 Graphing Trig Functions

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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# $ " %

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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

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Graphing Trig Functions