5.4 Sum and Difference Formulas
Formulas are on your equation
sheet and in book.
Find the exact value of cos 750.
cos 750 = cos(300 + 450)
= cos 300 cos 450 - sin 300 sin 450
3& 2# 1& 2#
$
!' $
!
=
$
!
$
2 % 2 " 2 % 2 !"
6! 2
=
4
Find the exact value of tan
!
! !
tan = tan( " )
12
4 6
!
!
!
12
Mult. By
3/3
3
tan " tan
1!
3
&
#
4
6
3
=
=
$
!
!
!
3
1 + tan tan
1 + (1)
3
4
6
%
"
3
3! 3 & 3' 3 # 9!6 3 +3
$
! =
=
$
!
9!3
3
'
3
3+ 3 %
"
12 ! 6 3
=
= 2! 3
6
Find the exact value of sin 420 cos 120 - cos 420 sin 120
sin 420 cos 120 - cos 420 sin 120 = sin(420 - 120)
= sin 300
1
=
2
Evaluate
cos(arctan 1 + arccos x)
This expression fits cos ( u + v )
u = arctan 1
and
v = arccos x
cos ( u + v ) = cos(arctan 1) cos(arccos x) sin(arctan 1) sin(arccos x)
2
2
2
=
x!
1! x
2
2
=
(
2 x!
1! x
2
2
)
&(
#
Pr ove : cos$ ' x ! = sin x
%2
"
!
!
'!
$
cos% ( x " = cos cos x + sin sin x
2
2
&2
#
= (0) cos x + (1) sin x
= sin x
Simplify:
3( #
&
cos$) ' !
2 "
%
3( #
&
3!
3!
cos$) ' ! = cos" cos + sin " sin
2
2
2 "
%
= (cos" )(0) + (sin " )(!1)
= " sin !
Find all solutions of
($
($
'
'
sin % x + " + sin % x ! " = !1
4#
4#
&
&
in the interval
[0,2! )
"
"
"
"
sin x cos + cos x sin + sin x cos ! cos x sin = !1
4
4
4
4
"
2 sin x cos = !1
4
1
sin x = !
2
' 2$
" = !1
2 sin x%%
"
2
&
#
5!
7!
x=
,x =
4
4