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5.5 Multiple-Angle and Product-Sum Formulas
Find all solutions in
(0,2! ) 2 cos x + sin 2 x = 0
2 cos x + 2 sin x cos x = 0
2 cos x(1 + sin x) = 0
cos x = 0
sin x = !1
! 3!
x= ,
2 2
3!
x=
2
Sketch the graph of y = 4 cos2 x - 2 over
[0,2! ]
= 2(2 cos2 x - 1)
= 2 cos 2x
2
!
!2
2!
Use the fact that
5 3!
cos x = ,
< x < 2!
13 2
to find sin 2x, cos 2x, and tan 2x
sin 2x = 2 sin x cos x =
120
' 12 $' 5 $
2% ! "% " = !
169
& 13 #& 13 #
cos 2x = 2 cos2 x - 1 =
119
' 25 $
2%
" !1 = !
169
& 169 #
tan 2x =
120
!
sin 2 x
120
169
=
=
cos 2 x ! 119 119
169
5
-12
13
Express sin 3x in terms of sin x.
sin 3x = sin (2x + x)
= sin 2x cos x + cos 2x sin x
= 2 sin x cos x cos x + (1 - 2 sin2 x) sin x
= 2 sin x cos2 x + sin x - 2 sin3 x
= 2 sin x(1 - sin2 x) + sin x - 2 sin3 x
= 2 sin x - 2 sin3 x + sin x - 2 sin3 x
= 3 sin x - 4 sin3 x
Rewrite sin4 x as a sum involving first powers of cos x
sin4 x = (sin2x)2 = &$ 1 ' cos 2 x #!
2
%
"
Foil
2
1
= 1 ! 2 cos 2 x + cos 2 2 x
4
(
)
1&
1 + cos 4 x #
= $1 ' 2 cos 2 x +
!
4%
2
"
1 1
1 1
= ! cos 2 x + + cos 4 x
4 2
8 8
3 1
1
= ! cos 2 x + cos 4 x
8 2
8
1
= (3 ! 4 cos 2 x + cos 4 x )
8
Find the exact value of sin 105o
sin 105o =
210 1 ! cos 210o
sin
=
2
2
3
1+
2
=
2
&2#
$ !
%2"
2+ 3
=
2
sin of 105 is positive
Writing Products as Sums
Rewrite
cos 5x sin 4x
cos 5x sin 4x =
as a sum or difference.
1
[sin(5 x + 4 x) ! sin(5 x ! 4 x)]
2
1
1
= sin 9 x ! sin x
2
2
Using a Sum-to-Product Formula
Find the exact value of
cos 1950 + cos 1050
cos 1950 + cos 1050
& 1950 + 1050 # & 1950 ' 1050 #
!! cos$$
!!
= 2 cos$$
2
2
%
" %
"
= 2 cos 1500 cos 450
&
3 #& 2 #
!$
!
= 2$$ '
!$ 2 !
2
%
"%
"
6
=!
2
Solving a Trigonometric Equation
Find all solutions of sin 5x + sin 3x = 0
in
[0,2! )
& 5 x + 3x # & 5 x ' 3x #
2 sin $
! cos$
!=0
% 2 " % 2 "
2sin 4x cos x = 0
sin 4x = 0
4 x = 0 + 2k!
4 x = ! + 2k!
! k!
x= +
4 2
k!
x=
2
cos x = 0
! 3!
"x = ,
2 2
Plug in 0, 1, 2, and 3
! ! 3!
5! 3! 7!
" x = 0, , , , ! , , ,
4 2 4
4 2 4