Download Module`4`Lesson`4:``Evaluating`Trig`Functions`

Module'4'Lesson'4:''Evaluating'Trig'Functions'
!
Recall!your!knowledge!of!the!Unit!Circle,!specifically!with!radian!measures.!
Note:!!These!specific!
ordered!pairs!are!
only!unique!to!the!
Unit!Circle!–!a!circle!
whose!radius!is!1.!
You!might!be!wondering!where!sine,!cosine,!and!tangent!fall!into!place!here.!!!
!
!
Evaluating'Sine'and'Cosine'
Each!ordered!pair!(x,!y)!corresponds!to!the!sine!and!cosine!value!of!an!angle!θ.!
!
!, ! = (cos ! , sin !)!
Tip:!!Since!(x,!y)!
!
appear!in!
So!cos ! = !!and!sin ! = !.!
alphabetical!order,!
!
so!do!(cosθ,!sinθ)!
1"
Example'1.!!Evaluate!cos
!!
!
!and!sin
!!
!
.!!!
(These!are!read!as!“cosine!of!2pi!over!3”!and!“sine!of!2pi!over!3”.)!
!!
Find!
!
!
!on!the!Unit!Circle.!!It!has!the!corresponding!ordered!pair! − ,
.!!
!
! !
2!
1
cos
=− !
3
2
and!
sin
2!
3
= !
3
2
!
Example'2.''Evaluate!cos
!!
!
!and!sin
!!
!
.!
(These!are!read!as!“cosine!of!7pi!over!4”!and!“sine!of!7pi!over!4.)!
!!
Find!
!
!on!the!Unit!Circle.!!It!has!the!corresponding!ordered!pair!
cos
!
!
,−
!
!
7!
2
= !
4
2
and!
sin
7!
2
=− !
4
2
Evaluating'Tangent'
!"# !
!
Recall!the!identity!that!tan ! = !"# !.!!Therefore,!tan ! = ! .!
'
!
Example'3.''Evaluate!tan .!
!
!
!
! !
Find! !on!the!Unit!Circle.!!It!has!the!corresponding!ordered!pair! ! , .!!!
!
! !
!
1
! sin 6
1
tan =
= 2 = !
!
6 cos
3
3
6
2
!
We!should!rationalize…!
1
!
3
∙
3
3
=
3
!
3
So!we!have!that!tan
2"
!
!
=
!
!
.!
.!!!
!
!
Example'4.''Evalute!tan − ! .!!!
!
Find!– !on!the!Unit!Circle.!!It!has!the!corresponding!ordered!pair!(0,!T1).!!!
!
!
sin − 2
!
1
tan −
=
=
−
= !"#$%&"$#!
!
2
0
cos − 2
!
Example'5.''Evalute!tan
!!
Find!
!!
!
.!!!
!on!the!Unit!Circle.!!It!has!the!corresponding!ordered!pair! −
!
3! − 2
3! sin 4
tan
=
= 2 = −1!
3!
4
2
cos 4
2
!
!
!
!
!
,
!
!
Positive'Trig'Values'by'Quadrant'
"
It!is!helpful!to!recognize!that!certain!trigonometric!values!are!positive!or!negative!
depending!on!the!quadrant!in!which!the!terminal!side!of!the!angle!lies.!!!
•
•
•
3"
!
Since!cos ! = !,!!"#$%&!values!are!positive!in!Quadrants!I!and!IV.!
Since!sin ! = !,!!"#!!values!are!positive!in!Quadrants!I!and!II.!
!
Since!tan ! = ! ,!!"#$%#!!values!are!positive!where!either!both!x!and!y!are!
positive!or!both!x!and!y!are!negative.!!This!occurs!in!Quadrants!I!and!III.!
.!!!
!
A!helpful!pneumonic!device!is!“A!Smart!Trig!Class”.!
!
!
"
4"
•
•
•
•
The!“A”!stands!for!“All!are!positive”.!
The!“S”!stands!for!“Sine!is!positive”.!
The!“T”!stands!for!“Tangent!is!positive”.!
The!“C”!stands!for!“Cosine!is!positive”.!
'
!
Evaluating'Secant,'Cosecant,'and'Cotangent'
!
Recall!the!“reciprocal”!properties!of!secant,!cosecant,!and!cotangent!from!Module!4!
Lesson!1.!!Consider!their!application!here!with!the!Unit!Circle.!
!
…And'therefore'we'use'these'
Reciprocal'Identities'
properties'to'evaluate'trig'
functions'using'the'Unit'Circle.'
sec ! =
1
!
cos !
!"# ! = !!!
(since!cos ! = !)!
csc ! =
1
!
sin !
!"! ! = !!!
!
!
(since!sin ! = !)!
1
cot ! = tan !!and!therefore!
!
!"# ! = !!!
cos !
!
!!
Example'6.''Evaluate!sec ! .!
!
Since!cos
!!
!
!
= !we!can!find!secant!by…!
!
sec
!
Example'7.!!Evaluate!csc −
Since!sin −
!
!!
!
=
!
!
!!
!
5! 1
= = 2!
1
3
2
.!
!we!can!find!cosecant!by…!
csc −
5!
1
2
=
= !
4
2
2
2
!
Always!rationalize!if!there!is!a!square!root!in!the!denominator.!
2
2 2 2
∙
=
= 2!
2
2 2
!
Therefore!csc −
'
5"
'
!
(since!tan ! = ! )!
cot ! = sin ! !
!
!!
!
= 2.!
Example'8.!!Evaluate!cot !.!
!
!
!
!
!
Since!tan ! = ! = !!we!can!find!cotangent!by!finding!the!reciprocal!of!! .!
!
! 0
cot = = 0!
2 1
6"