SECTION 6.6 Product-to-Sum and Sum-to-Product Formulas
EXAMPLE 2
495
Expressing Sums (or Differences) as a Product
y sines or cosines.
50)
Express each sum or difference as a product of sines and/or cosines,
(a) sin(50) - sin (30)
Solution
(b) cos(30) + cos(20)
(a) We use formula (7) to get
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. . . . , . . , . . . . •
si n(jo ) — sm^jp )
^ sin
2
2
= 2 sin 0 cos(4(0
(b) cos(30) + cos(20 )x
z.•, COS 3e +
O/3
Z.17
'i/J
O"
O/3
Z.C7
HJ»
Formula (&)
50
0
= 2 cos — cos /
sin(20)J
WORK
4
PROBLEM
11.
• 16.6 Assess Your Understanding
Skill Building
In Problems 1-10, express each product as a sum containing only sines or cosines.
\. sin(40) sin(20)
2. cos(40) cos(20)
3. sin(40) cos(20)
4. sin(30) sin(50)
6. sin(40) cos(60)
6
30
0
9. sin — cos —
8. cos (30) cos (40)
7. sin0sin(20)
5. cos(30) cos(50)
10. sin-
In Problems 11-18, express each sum or difference as a product of sines and/or cosines.
sin(40) - sin(20)
12. sin(40) + sin(20)
13. cos(20) + cos(40)
w
15. sin0 + sin(30)
16.
COS0 -}-
0
cos(30)
50
COSy
14. cos(50) - cos(30)
0
30
17. cos — — cos —
30
18. sin — - smy
In Problems 19-36, establish each identity.
19.
22.
sin0 + sin(30)
2sin(20)
COS0 - cos(30)
sin(30) — sin 0
in
)
""
23
(20)
COS0 +
COS0 -
sin(40) + sin(80)
,.nlf*Q\n
) through (9)
10
11
sin a + sin B
sin a - sin B
11 sin a + sin B
cos a + cos B
1A
cos 0 - cos(50)
sm0 + sin(50)
sin(40) - sin(80)
(60)
2
a + B
2
35. 1 + cos(20) + cos(40) + cos(60) = 4 cos 0 cos(20) cos(30)
36. 1 - cos(20) + cos(40) - cos(60) = 4 sin 0 cos (20) sin(30)
l(\Q\
' cos(40) - cos(80)
°0t
cos (40) - cos(80)
cos(40) + cos(80)
Un(i
•2A
a — )3
1(20)
ta
26. sin0[sin(30) + sin(50)] = cos 0[cos(30) - cos(50)]
tan(20)
a + B
tan(30)
' cos(40) + cos(20)
cos (30)
sin(40) + sin(80)
sin(40) - sin(80)
= COS0
sin 0 + sin(30)
25. sin0[sin0 + sin(30)] = cos0[cos0 - cos(30)]
cos (40) + cos(80)
sin(40) + sin(20)
cos (30)
2 cos(20)
I')
JZ.
cos a + COS B
a
cos a - cos B
74
sin a — sin B
cos a - cos B
C0t
a- B
a + B
cot
T2
a + )3
2