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A2HCh0705 Multiple Angles and Product to Sum Formulas
Homework and Reading
Read p569– 576
(1) HW p577 #1– 53 odd
(2) HW p578 #55 – 81 odd
#95–109 odd, 121
Goal
Multiple-Angles Formulas
Students use multiple-angle formulas, power-reducing formulas, half-angle formulas, productto-sum formulas, and sum-to-product formulas to rewrite and evaluate trigonometric
functions,
Graph y = cos 4 x ! sin 4 x on [ 0, 2" )
Example
Multiple – Angles Formula
Double – Angle Formulas
cos 2u = cos 2 u ! sin 2 u
= 2 cos 2 u ! 1
2 tan u
tan 2u =
1 ! tan 2 u
Example
Simplify cos x ! sin x
4
sin 2u = 2 sin u cosu
(
4
2
= cos 2x
1
2
" 5"
x= ,
6 6
solve
-!
2 tan x
= 2 cos x
1 ! tan 2 x
2 tan x = 2 cos x 1 ! tan 2 x
)
)
sec x 2 sin 2
multiply
sin x =
simplify
No Solution
)
factor
sin 2x = 0
2x = " n
"n
x=
2
"
3"
x = 0, , " ,
2
2
2 sin x ! 1 = 0
2
2
" 3" 5" 7"
x= ,
,
,
4 4 4 4
solve
(
cos 4 x = cos 2 x
1 + cos 2u
cos u =
2
2
1 ! cos 2u
tan u =
1 + cos 2u
2
u
1 ! cos 2u
=±
2
2
u 1 ! cos 2u
sin u
tan =
=
2
sin u
1 + cos 2u
and cos
u
2
cos
3"
2
set to zero
cos 2x = !1
2x = " + 2" n
"
+ "n
2
" 3"
x= ,
2 2
x=
2
rewrite
(1 + cos 2x )
2
rewrite
2
22
1
2
= (1 + cos 2x )
4
1
= 1 + 2 cos 2x + cos 2 2x
4
1!
1 + cos 4x $
= # 1 + 2 cos 2x +
4"
2
%&
u
1 + cos 2u
=±
2
2
depened on the quadrant in which
)
! 1 + cos 2x $
=#
&%
2
"
=
Half – Angle Formulas
sin
simplify
rewrite
Rewrite cos 4 x as a sum of first powers of cosine.
1 ! cos 2u
sin u =
2
2
x=
Example
More Formulas
Power – Reducing Formulas
rewrite
sin x = !1
2 sin 2x ( cos 2x + 1) = 0
cos 2x + 1 = 0
2 sin 2x = 0
2
sin x = ±
" 5"
,
6 6
1
2
sin 4x + 2 sin 2x = 0
2 sin 2x cos 2x + 2 sin 2x = 0
set to zero
cos x 2 sin x ! 1 = 0
factor
Try : Solve sin 4x = !2 sin 2x on [ 0, 2" )
rewrite
2 sin 2 x cos x ! cos x = 0
u
x=
set to zero
2 sin x cos x ( sin x ) = cos x
2
3!
)
(
))
x ! sin x ! 1) = 0
rewrite
Use sin 2x = 2 sin x cos x rewrite and substitute
The signs of sin
2!
sec x ( 2 sin x ! 1) ( sin x + 1) = 0 factor
sec x = 0
2 sin x ! 1 = 0 sin x + 1 = 0
divide by 2
Try : Solve sin 2x sin x = cos x
" 3"
,
2 2
(
multiply
sin x cos 2 x
sin 2 x
=
! cos x
cos x cos x
cos 2 x
sin x cos 2 x sin 2 x
=
!
cos x cos x cos x
2
sin x sin x cos 2 x
+
!
=0
cos x cos x cos x
x=
(
(
rewrite
tan x = cos x ! cos x tan 2 x
cos x = 0
!
1
sin x + sin 2 x ! cos 2 x = 0
cos x
sec x sin x + sin 2 x ! 1 ! sin 2 x = 0
Move 2 cos x and use tan 2x formula
(
y = cos 2x
-1
Solve tan 2x ! 2 cos x = 0
(
graph this function
y = cos x
0
Example
tan x = cos x 1 ! tan 2 x
1
2!
Period :
=!
2
sin x =
2
)) rewrite
2
Use cos 2x = 2 cos x ! 1
rewrite
cos x ( 2 sin x ! 1) = 0 factor
cos x = 0 2 sin x ! 1 = 0
(
(
) factor
1 • cos x + cos x ! 1
2
= 2 cos 2 x ! 1
Solve sin 2x ! cos x = 0
" 3"
x= ,
2 2
)(
(
=
= 1 ! 2 sin u
2 sin x cos x ! cos x = 0
Half-Angle Formulas
4
cos x ! sin x = cos 2 x + sin 2 x cos 2 x ! sin 2 x
4
2
Use sin 2x = 2 sin x cos x rewrite and substitute
Power-Reducing Formulas
p1
(
u
2
lies.
=
)
1 ! 2 4 cos 2x 1 cos 4x $
+
+ +
4 #" 2
2
2
2 &%
1
= ( 3 + 4 cos 2x + cos 4x )
8
power rule
rewrite
multiply
rewrite
rewrite
rewrite
factor
solve
A2HCh0705 Multiple Angles and Product to Sum Formulas
more examples
Example
Find the exact values of cos 165º .
cos 165º is in quadrant II , use negative
cos 165º = cos u 2 , so u = 330º
1 + cos 330º
2
cos 165º = !
rewrite
1+ 3 2
=!
2
Product-to-Sum Formulas
simplify
More Formulas
simplify
simplify
solve
solve
Sum – to – Product Formulas
1
sin u sin v = "# cos ( u ! v ) ! cos ( u + v ) $%
2
1
cosu cos v = "# cos ( u ! v ) + cos ( u + v ) $%
2
1
sin u cos v = "#sin ( u + v ) + sin ( u ! v ) $%
2
1
cosu sin v = "#sin ( u + v ) ! sin ( u ! v ) $%
2
Example
replace by using the formula
" 195º !105º % " 195º !105º %
sin195º ! sin105º = 2 cos $
'& sin $#
'&
2
2
#
= 2 cos (150º ) sin ( 45º )
rewrite
"
3% " 2 %
= 2$ !
'$ 2 '
2
#
&#
&
simplify
=!
Example
Verify the identity
6
2
simplify
sin 7x ! sin 5x
= ! cot 6x
cos 7x ! cos 5x
replace by using the formula
" 7x + 5x % " 7x ! 5x %
2 cos $
sin
2 '& $#
2 '&
#
sin 7x ! sin 5x
=
cos 7x ! cos 5x
" 7x + 5x % " 7x ! 5x %
!2 sin $
sin
2 '& $#
2 '&
#
" 12x %
cos $
# 2 '&
=
" 12x %
! sin $
# 2 '&
simplify
cos ( 6x )
=
! sin ( 6x )
= ! cot 6x
simplify
rewrite
Try : Verify the identity cos 4 x ! sin 4 x = cos 2x
cos 4 x ! sin 4 x = cos 2x
2
2
cos x + sin x cos 2 x ! sin 2 x = factor
)(
(
)
)
1 cos 2 x ! sin 2 x =
cos 2x =
! u + v$
! u ' v$
sin u + sin v = 2 sin #
cos #
" 2 &%
" 2 &%
! u + v$ ! u ' v$
sin u ' sin v = 2 cos #
sin
" 2 &% #" 2 &%
! u + v$
! u ' v$
cosu + cos v = 2 cos #
cos #
" 2 &%
" 2 &%
! u + v$ ! u ' v$
cosu ' cos v = '2 sin #
sin
" 2 &% #" 2 &%
Example
Find the exact value of sin195º ! sin105º
(
square / simplify
More Formulas
Product – to – Sum Formulas
Sum-to-Product Formulas
# 1 " cos x &
2%
= cos x
2 ('
$
1 " cos x = cos x
1 = 2 cos x
1
= cos x
2
! 5!
x= ,
3 3
rewrite
2+ 3
2
=!
p2
x
Solve 2 sin 2 = cos x on [ 0, 2! )
2
x
2 sin 2 = cos x
2
2
# 1 " cos x &
2% ±
replace sin with formula, use ±
( = cos x
2
$
'
Example
simplify
rewrite / Double Angle
Solve cos 3x + cos x = 0 on [ 0, 2! )
cos 3x + cos x = 0
" 3x + x %
" 3x ( x %
2 cos $
cos $
=0
'
# 2 &
# 2 '&
2 cos ( 2x ) cos ( x ) = 0
replace by using the formula
simplify
cos ( 2x ) cos ( x ) = 0
simplify
cos x = 0
cos 2x = 0
!
!
x = + 2! n
2x = + 2! n
2
2
or
!
x = + !n
3!
4
x=
+ 2! n
or
2
3!
2x =
+ 2! n
2
3!
x=
+ !n
! 3!
4
! 3! 5! 7! x = 2 , 2
x= ,
,
,
4 4 4 4
solve each
Try : Find the exact values of cos120º + cos 30º
replace by using the formula
! 120º +30º $
! 120º '30º $
cos120º + cos 30º = 2 cos #
cos #
2
2
"
%&
"
%&
! 150º $
! 90º $
= 2 cos #
cos #
" 2 %&
" 2 %&
simplify
= 2 cos ( 45º +30º ) cos ( 45º )
rewrite
= 2 cos ( 75º ) cos ( 45º )
simplify
= 2 ( cos 45º cos 30º ' sin 45º sin 30º ) cos ( 45º )
rewrite
! 2
3
2 1$
2
= 2#
•
'
•
•
2
2 2 &% 2
" 2
simplify
! 6 ' 2$
2
12 ' 4 2 3 ' 2
= 2#
=
&• 2 =
4
4
4
"
%
simplify