1. Products to Sums

2. Products to Sums
2sin A cos B  sin  A  B   sin  A  B 

3. Products to Sums
2sin A cos B  sin  A  B   sin  A  B 
2cos A cos B  cos  A  B   cos  A  B 

4. Products to Sums
2sin A cos B  sin  A  B   sin  A  B 
2cos A cos B  cos  A  B   cos  A  B 
2sin A sin B  cos  A  B   cos  A  B 

5. Products to Sums
2sin A cos B  sin  A  B   sin  A  B 
2cos A cos B  cos  A  B   cos  A  B 
2sin A sin B  cos  A  B   cos  A  B 
eg (i) Express as a sum or difference of trig functions

6. Products to Sums
2sin A cos B  sin  A  B   sin  A  B 
2cos A cos B  cos  A  B   cos  A  B 
2sin A sin B  cos  A  B   cos  A  B 
eg (i) Express as a sum or difference of trig functions
a ) 2cos5 x sin x

7. Products to Sums
2sin A cos B  sin  A  B   sin  A  B 
2cos A cos B  cos  A  B   cos  A  B 
2sin A sin B  cos  A  B   cos  A  B 
eg (i) Express as a sum or difference of trig functions
a ) 2cos5 x sin x  2sin x cos5 x

8. Products to Sums
2sin A cos B  sin  A  B   sin  A  B 
2cos A cos B  cos  A  B   cos  A  B 
2sin A sin B  cos  A  B   cos  A  B 
eg (i) Express as a sum or difference of trig functions
a ) 2cos5 x sin x  2sin x cos5 x
 sin  x  5 x   sin  x  5 x 

9. Products to Sums
2sin A cos B  sin  A  B   sin  A  B 
2cos A cos B  cos  A  B   cos  A  B 
2sin A sin B  cos  A  B   cos  A  B 
eg (i) Express as a sum or difference of trig functions
a ) 2cos5 x sin x  2sin x cos5 x
 sin  x  5 x   sin  x  5 x 
 sin 6 x  sin  4 x 

10. Products to Sums
2sin A cos B  sin  A  B   sin  A  B 
2cos A cos B  cos  A  B   cos  A  B 
2sin A sin B  cos  A  B   cos  A  B 
eg (i) Express as a sum or difference of trig functions
a ) 2cos5 x sin x  2sin x cos5 x
 sin  x  5 x   sin  x  5 x 
 sin 6 x  sin  4 x 
 sin 6 x  sin 4 x

11. Products to Sums
2sin A cos B  sin  A  B   sin  A  B 
2cos A cos B  cos  A  B   cos  A  B 
2sin A sin B  cos  A  B   cos  A  B 
eg (i) Express as a sum or difference of trig functions
a ) 2cos5 x sin x  2sin x cos5 x
 sin  x  5 x   sin  x  5 x 
 sin 6 x  sin  4 x 
 sin 6 x  sin 4 x
b) cos3 cos5

12. Products to Sums
2sin A cos B  sin  A  B   sin  A  B 
2cos A cos B  cos  A  B   cos  A  B 
2sin A sin B  cos  A  B   cos  A  B 
eg (i) Express as a sum or difference of trig functions
a ) 2cos5 x sin x  2sin x cos5 x
 sin  x  5 x   sin  x  5 x 
 sin 6 x  sin  4 x 
 sin 6 x  sin 4 x
1
b) cos3 cos5   2cos3 cos5 
2

13. Products to Sums
2sin A cos B  sin  A  B   sin  A  B 
2cos A cos B  cos  A  B   cos  A  B 
2sin A sin B  cos  A  B   cos  A  B 
eg (i) Express as a sum or difference of trig functions
a ) 2cos5 x sin x  2sin x cos5 x
 sin  x  5 x   sin  x  5 x 
 sin 6 x  sin  4 x 
 sin 6 x  sin 4 x
1
b) cos3 cos5   2cos3 cos5 
2
1
  cos  3  5   cos  3  5  
2

14. Products to Sums
2sin A cos B  sin  A  B   sin  A  B 
2cos A cos B  cos  A  B   cos  A  B 
2sin A sin B  cos  A  B   cos  A  B 
eg (i) Express as a sum or difference of trig functions
a ) 2cos5 x sin x  2sin x cos5 x
 sin  x  5 x   sin  x  5 x 
 sin 6 x  sin  4 x 
 sin 6 x  sin 4 x
1
b) cos3 cos5   2cos3 cos5 
2
1
  cos  3  5   cos  3  5  
2
1
  cos8  cos  2  
2

15. Products to Sums
2sin A cos B  sin  A  B   sin  A  B 
2cos A cos B  cos  A  B   cos  A  B 
2sin A sin B  cos  A  B   cos  A  B 
eg (i) Express as a sum or difference of trig functions
a ) 2cos5 x sin x  2sin x cos5 x
 sin  x  5 x   sin  x  5 x 
 sin 6 x  sin  4 x 
 sin 6 x  sin 4 x
1
b) cos3 cos5   2cos3 cos5 
2
1
  cos  3  5   cos  3  5  
2
1 1
  cos8  cos  2     cos8  cos 2 
2 2

16. (ii) Evaluate 2sin 45 cos15

17. (ii) Evaluate 2sin 45 cos15  sin  45  15   sin  45  15 

18. (ii) Evaluate 2sin 45 cos15  sin  45  15   sin  45  15 
 sin 60  sin 30

19. (ii) Evaluate 2sin 45 cos15  sin  45  15   sin  45  15 
 sin 60  sin 30
3 1
 
2 2

20. (ii) Evaluate 2sin 45 cos15  sin  45  15   sin  45  15 
 sin 60  sin 30
3 1
 
2 2
3 1

2

21. (ii) Evaluate 2sin 45 cos15  sin  45  15   sin  45  15 
 sin 60  sin 30
3 1
 
2 2
3 1

2
Sums to Products

22. (ii) Evaluate 2sin 45 cos15  sin  45  15   sin  45  15 
 sin 60  sin 30
3 1
 
2 2
3 1

2
Sums to Products
1 1
sin A  sin B  2sin  A  B  cos  A  B 
2 2

23. (ii) Evaluate 2sin 45 cos15  sin  45  15   sin  45  15 
 sin 60  sin 30
3 1
 
2 2
3 1

2
Sums to Products
1 1
sin A  sin B  2sin  A  B  cos  A  B  (2 sine half sum cos half diff)
2 2

24. (ii) Evaluate 2sin 45 cos15  sin  45  15   sin  45  15 
 sin 60  sin 30
3 1
 
2 2
3 1

2
Sums to Products
1 1
sin A  sin B  2sin  A  B  cos  A  B  (2 sine half sum cos half diff)
2 2
1 1
cos A  cos B  2cos  A  B  cos  A  B 
2 2

25. (ii) Evaluate 2sin 45 cos15  sin  45  15   sin  45  15 
 sin 60  sin 30
3 1
 
2 2
3 1

2
Sums to Products
1 1
sin A  sin B  2sin  A  B  cos  A  B  (2 sine half sum cos half diff)
2 2
1 1
cos A  cos B  2cos  A  B  cos  A  B  (2 cos half sum cos half diff)
2 2

26. (ii) Evaluate 2sin 45 cos15  sin  45  15   sin  45  15 
 sin 60  sin 30
3 1
 
2 2
3 1

2
Sums to Products
1 1
sin A  sin B  2sin  A  B  cos  A  B  (2 sine half sum cos half diff)
2 2
1 1
cos A  cos B  2cos  A  B  cos  A  B  (2 cos half sum cos half diff)
2 2
1 1
cos A  cos B  2sin  A  B  sin  A  B 
2 2

27. (ii) Evaluate 2sin 45 cos15  sin  45  15   sin  45  15 
 sin 60  sin 30
3 1
 
2 2
3 1

2
Sums to Products
1 1
sin A  sin B  2sin  A  B  cos  A  B  (2 sine half sum cos half diff)
2 2
1 1
cos A  cos B  2cos  A  B  cos  A  B  (2 cos half sum cos half diff)
2 2
1 1
cos A  cos B  2sin  A  B  sin  A  B  (minus 2 sine half sum
2 2 sine half diff)

28. eg (i) Convert into products of trig functions

29. eg (i) Convert into products of trig functions
a ) cos3 A  cos5 A

30. eg (i) Convert into products of trig functions
1 1
a ) cos3 A  cos5 A  2sin  8 A  sin  2 A 
2 2

31. eg (i) Convert into products of trig functions
1 1
a ) cos3 A  cos5 A  2sin  8 A  sin  2 A 
2 2
 2sin 4 A sin   A 

32. eg (i) Convert into products of trig functions
1 1
a ) cos3 A  cos5 A  2sin  8 A  sin  2 A 
2 2
 2sin 4 A sin   A 
 2sin 4 A sin A

33. eg (i) Convert into products of trig functions
1 1
a ) cos3 A  cos5 A  2sin  8 A  sin  2 A 
2 2
 2sin 4 A sin   A 
 2sin 4 A sin A
b) sin 6 x  sin 4 x

34. eg (i) Convert into products of trig functions
1 1
a ) cos3 A  cos5 A  2sin  8 A  sin  2 A 
2 2
 2sin 4 A sin   A 
 2sin 4 A sin A
b) sin 6 x  sin 4 x  sin 6 x  sin  4 x 

35. eg (i) Convert into products of trig functions
1 1
a ) cos3 A  cos5 A  2sin  8 A  sin  2 A 
2 2
 2sin 4 A sin   A 
 2sin 4 A sin A
b) sin 6 x  sin 4 x  sin 6 x  sin  4 x 
1 1
 2sin  2 x  cos 10 x 
2 2

36. eg (i) Convert into products of trig functions
1 1
a ) cos3 A  cos5 A  2sin  8 A  sin  2 A 
2 2
 2sin 4 A sin   A 
 2sin 4 A sin A
b) sin 6 x  sin 4 x  sin 6 x  sin  4 x 
1 1
 2sin  2 x  cos 10 x 
2 2
 2sin x cos5 x

37. eg (i) Convert into products of trig functions
1 1
a ) cos3 A  cos5 A  2sin  8 A  sin  2 A 
2 2
 2sin 4 A sin   A 
 2sin 4 A sin A
b) sin 6 x  sin 4 x  sin 6 x  sin  4 x 
1 1
 2sin  2 x  cos 10 x 
2 2
 2sin x cos5 x
(ii) Solve sin x  sin 3 x  0 0  x  360

38. eg (i) Convert into products of trig functions
1 1
a ) cos3 A  cos5 A  2sin  8 A  sin  2 A 
2 2
 2sin 4 A sin   A 
 2sin 4 A sin A
b) sin 6 x  sin 4 x  sin 6 x  sin  4 x 
1 1
 2sin  2 x  cos 10 x 
2 2
 2sin x cos5 x
(ii) Solve sin x  sin 3 x  0 0  x  360
2sin 2 x cos   x   0

39. eg (i) Convert into products of trig functions
1 1
a ) cos3 A  cos5 A  2sin  8 A  sin  2 A 
2 2
 2sin 4 A sin   A 
 2sin 4 A sin A
b) sin 6 x  sin 4 x  sin 6 x  sin  4 x 
1 1
 2sin  2 x  cos 10 x 
2 2
 2sin x cos5 x
(ii) Solve sin x  sin 3 x  0 0  x  360
2sin 2 x cos   x   0
2sin 2 x cos x  0

40. eg (i) Convert into products of trig functions
1 1
a ) cos3 A  cos5 A  2sin  8 A  sin  2 A 
2 2
 2sin 4 A sin   A 
 2sin 4 A sin A
b) sin 6 x  sin 4 x  sin 6 x  sin  4 x 
1 1
 2sin  2 x  cos 10 x 
2 2
 2sin x cos5 x
(ii) Solve sin x  sin 3 x  0 0  x  360
2sin 2 x cos   x   0
2sin 2 x cos x  0
sin 2 x  0 or cos x  0

41. eg (i) Convert into products of trig functions
1 1
a ) cos3 A  cos5 A  2sin  8 A  sin  2 A 
2 2
 2sin 4 A sin   A 
 2sin 4 A sin A
b) sin 6 x  sin 4 x  sin 6 x  sin  4 x 
1 1
 2sin  2 x  cos 10 x 
2 2
 2sin x cos5 x
(ii) Solve sin x  sin 3 x  0 0  x  360
2sin 2 x cos   x   0
2sin 2 x cos x  0
sin 2 x  0 or cos x  0
2 x  0 ,180 ,360 ,540 ,720

42. eg (i) Convert into products of trig functions
1 1
a ) cos3 A  cos5 A  2sin  8 A  sin  2 A 
2 2
 2sin 4 A sin   A 
 2sin 4 A sin A
b) sin 6 x  sin 4 x  sin 6 x  sin  4 x 
1 1
 2sin  2 x  cos 10 x 
2 2
 2sin x cos5 x
(ii) Solve sin x  sin 3 x  0 0  x  360
2sin 2 x cos   x   0
2sin 2 x cos x  0
sin 2 x  0 or cos x  0
2 x  0 ,180 ,360 ,540 ,720
x  0 ,90 ,180 , 270 ,360

43. eg (i) Convert into products of trig functions
1 1
a ) cos3 A  cos5 A  2sin  8 A  sin  2 A 
2 2
 2sin 4 A sin   A 
 2sin 4 A sin A
b) sin 6 x  sin 4 x  sin 6 x  sin  4 x 
1 1
 2sin  2 x  cos 10 x 
2 2
 2sin x cos5 x
(ii) Solve sin x  sin 3 x  0 0  x  360
2sin 2 x cos   x   0
2sin 2 x cos x  0
sin 2 x  0 or cos x  0
2 x  0 ,180 ,360 ,540 ,720 x  90 , 270
x  0 ,90 ,180 , 270 ,360

44. eg (i) Convert into products of trig functions
1 1
a ) cos3 A  cos5 A  2sin  8 A  sin  2 A 
2 2
 2sin 4 A sin   A 
 2sin 4 A sin A
b) sin 6 x  sin 4 x  sin 6 x  sin  4 x 
1 1
 2sin  2 x  cos 10 x 
2 2
 2sin x cos5 x
(ii) Solve sin x  sin 3 x  0 0  x  360
2sin 2 x cos   x   0
2sin 2 x cos x  0
sin 2 x  0 or cos x  0
2 x  0 ,180 ,360 ,540 ,720 x  90 , 270
x  0 ,90 ,180 , 270 ,360  x  0 ,90 ,180 , 270 ,360

45. Exercise 2F; 1b, 2b, 3a, 9, 10ace