1. Products to Sums

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

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

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

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

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

7. Products to Sums
   2sin cos sin sinA B A B A B   
   2cos cos cos cosA B A B A B   
   2sin sin cos cosA B A B A B   
eg (i) Express as a sum or difference of trig functions
) 2cos5 sina x x 2sin cos5x x

8. Products to Sums
   2sin cos sin sinA B A B A B   
   2cos cos cos cosA B A B A B   
   2sin sin cos cosA B A B A B   
eg (i) Express as a sum or difference of trig functions
) 2cos5 sina x x 2sin cos5x x
   sin 5 sin 5x x x x   

9. Products to Sums
   2sin cos sin sinA B A B A B   
   2cos cos cos cosA B A B A B   
   2sin sin cos cosA B A B A B   
eg (i) Express as a sum or difference of trig functions
) 2cos5 sina x x 2sin cos5x x
   sin 5 sin 5x x x x   
 sin6 sin 4x x  

10. Products to Sums
   2sin cos sin sinA B A B A B   
   2cos cos cos cosA B A B A B   
   2sin sin cos cosA B A B A B   
eg (i) Express as a sum or difference of trig functions
) 2cos5 sina x x 2sin cos5x x
   sin 5 sin 5x x x x   
 sin6 sin 4x x  
sin6 sin 4x x 

11. Products to Sums
   2sin cos sin sinA B A B A B   
   2cos cos cos cosA B A B A B   
   2sin sin cos cosA B A B A B   
eg (i) Express as a sum or difference of trig functions
) 2cos5 sina x x 2sin cos5x x
   sin 5 sin 5x x x x   
 sin6 sin 4x x  
sin6 sin 4x x 
) cos3 cos5b  

12. Products to Sums
   2sin cos sin sinA B A B A B   
   2cos cos cos cosA B A B A B   
   2sin sin cos cosA B A B A B   
eg (i) Express as a sum or difference of trig functions
) 2cos5 sina x x 2sin cos5x x
   sin 5 sin 5x x x x   
 sin6 sin 4x x  
sin6 sin 4x x 
) cos3 cos5b    
1
2cos3 cos5
2
 

13. Products to Sums
   2sin cos sin sinA B A B A B   
   2cos cos cos cosA B A B A B   
   2sin sin cos cosA B A B A B   
eg (i) Express as a sum or difference of trig functions
) 2cos5 sina x x 2sin cos5x x
   sin 5 sin 5x x x x   
 sin6 sin 4x x  
sin6 sin 4x x 
) cos3 cos5b  
    
1
cos 3 5 cos 3 5
2
      
 
1
2cos3 cos5
2
 

14. Products to Sums
   2sin cos sin sinA B A B A B   
   2cos cos cos cosA B A B A B   
   2sin sin cos cosA B A B A B   
eg (i) Express as a sum or difference of trig functions
) 2cos5 sina x x 2sin cos5x x
   sin 5 sin 5x x x x   
 sin6 sin 4x x  
sin6 sin 4x x 
) cos3 cos5b  
    
1
cos 3 5 cos 3 5
2
      
  
1
cos8 cos 2
2
   
 
1
2cos3 cos5
2
 

15. Products to Sums
   2sin cos sin sinA B A B A B   
   2cos cos cos cosA B A B A B   
   2sin sin cos cosA B A B A B   
eg (i) Express as a sum or difference of trig functions
) 2cos5 sina x x 2sin cos5x x
   sin 5 sin 5x x x x   
 sin6 sin 4x x  
sin6 sin 4x x 
) cos3 cos5b  
    
1
cos 3 5 cos 3 5
2
      
  
1
cos8 cos 2
2
     
1
cos8 cos2
2
  
 
1
2cos3 cos5
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   
sin60 sin30  

19. (ii) Evaluate 2sin 45 cos15 
   sin 45 15 sin 45 15   
sin60 sin30  
3 1
2 2
 

20. (ii) Evaluate 2sin 45 cos15 
   sin 45 15 sin 45 15   
sin60 sin30  
3 1
2 2
 
3 1
2



21. (ii) Evaluate 2sin 45 cos15 
   sin 45 15 sin 45 15   
sin60 sin30  
3 1
2 2
 
3 1
2


Sums to Products

22. (ii) Evaluate 2sin 45 cos15 
   sin 45 15 sin 45 15   
sin60 sin30  
3 1
2 2
 
3 1
2


Sums to Products
   
1 1
sin sin 2sin cos
2 2
A B A B A B   

23. (ii) Evaluate 2sin 45 cos15 
   sin 45 15 sin 45 15   
sin60 sin30  
3 1
2 2
 
3 1
2


Sums to Products
   
1 1
sin sin 2sin cos
2 2
A B A B A B    (2 sine half sum cos half diff)

24. (ii) Evaluate 2sin 45 cos15 
   sin 45 15 sin 45 15   
sin60 sin30  
3 1
2 2
 
3 1
2


Sums to Products
   
1 1
sin sin 2sin cos
2 2
A B A B A B    (2 sine half sum cos half diff)
   
1 1
cos cos 2cos cos
2 2
A B A B A B   

25. (ii) Evaluate 2sin 45 cos15 
   sin 45 15 sin 45 15   
sin60 sin30  
3 1
2 2
 
3 1
2


Sums to Products
   
1 1
sin sin 2sin cos
2 2
A B A B A B    (2 sine half sum cos half diff)
   
1 1
cos cos 2cos cos
2 2
A B A B A B    (2 cos half sum cos half diff)

26. (ii) Evaluate 2sin 45 cos15 
   sin 45 15 sin 45 15   
sin60 sin30  
3 1
2 2
 
3 1
2


Sums to Products
   
1 1
sin sin 2sin cos
2 2
A B A B A B    (2 sine half sum cos half diff)
   
1 1
cos cos 2cos cos
2 2
A B A B A B    (2 cos half sum cos half diff)
   
1 1
cos cos 2sin sin
2 2
A B A B A B    

27. (ii) Evaluate 2sin 45 cos15 
   sin 45 15 sin 45 15   
sin60 sin30  
3 1
2 2
 
3 1
2


Sums to Products
   
1 1
sin sin 2sin cos
2 2
A B A B A B    (2 sine half sum cos half diff)
   
1 1
cos cos 2cos cos
2 2
A B A B A B    (2 cos half sum cos half diff)
   
1 1
cos cos 2sin sin
2 2
A B A B A B     (minus 2 sine half sum
sine half diff)

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

29. eg (i) Convert into products of trig functions
) cos3 cos5a A A

30. eg (i) Convert into products of trig functions
) cos3 cos5a A A    
1 1
2sin 8 sin 2
2 2
A A  

31. eg (i) Convert into products of trig functions
) cos3 cos5a A A    
1 1
2sin 8 sin 2
2 2
A A  
 2sin 4 sinA A  

32. eg (i) Convert into products of trig functions
) cos3 cos5a A A    
1 1
2sin 8 sin 2
2 2
A A  
 2sin 4 sinA A  
2sin 4 sinA A

33. eg (i) Convert into products of trig functions
) cos3 cos5a A A    
1 1
2sin 8 sin 2
2 2
A A  
 2sin 4 sinA A  
2sin 4 sinA A
) sin6 sin 4b x x

34. eg (i) Convert into products of trig functions
) cos3 cos5a A A    
1 1
2sin 8 sin 2
2 2
A A  
 2sin 4 sinA A  
2sin 4 sinA A
) sin6 sin 4b x x  sin6 sin 4x x  

35. eg (i) Convert into products of trig functions
) cos3 cos5a A A    
1 1
2sin 8 sin 2
2 2
A A  
 2sin 4 sinA A  
2sin 4 sinA A
) sin6 sin 4b x x  sin6 sin 4x x  
   
1 1
2sin 2 cos 10
2 2
x x

36. eg (i) Convert into products of trig functions
) cos3 cos5a A A    
1 1
2sin 8 sin 2
2 2
A A  
 2sin 4 sinA A  
2sin 4 sinA A
) sin6 sin 4b x x  sin6 sin 4x x  
   
1 1
2sin 2 cos 10
2 2
x x
2sin cos5x x

37. eg (i) Convert into products of trig functions
) cos3 cos5a A A    
1 1
2sin 8 sin 2
2 2
A A  
 2sin 4 sinA A  
2sin 4 sinA A
) sin6 sin 4b x x  sin6 sin 4x x  
   
1 1
2sin 2 cos 10
2 2
x x
2sin cos5x x
(ii) Solve sin sin3 0 0 360x x x    

38. eg (i) Convert into products of trig functions
) cos3 cos5a A A    
1 1
2sin 8 sin 2
2 2
A A  
 2sin 4 sinA A  
2sin 4 sinA A
) sin6 sin 4b x x  sin6 sin 4x x  
   
1 1
2sin 2 cos 10
2 2
x x
2sin cos5x x
(ii) Solve sin sin3 0 0 360x x x    
 2sin 2 cos 0x x 

39. eg (i) Convert into products of trig functions
) cos3 cos5a A A    
1 1
2sin 8 sin 2
2 2
A A  
 2sin 4 sinA A  
2sin 4 sinA A
) sin6 sin 4b x x  sin6 sin 4x x  
   
1 1
2sin 2 cos 10
2 2
x x
2sin cos5x x
(ii) Solve sin sin3 0 0 360x x x    
 2sin 2 cos 0x x 
2sin 2 cos 0x x 

40. eg (i) Convert into products of trig functions
) cos3 cos5a A A    
1 1
2sin 8 sin 2
2 2
A A  
 2sin 4 sinA A  
2sin 4 sinA A
) sin6 sin 4b x x  sin6 sin 4x x  
   
1 1
2sin 2 cos 10
2 2
x x
2sin cos5x x
(ii) Solve sin sin3 0 0 360x x x    
 2sin 2 cos 0x x 
2sin 2 cos 0x x 
sin 2 0 or cos 0x x 

41. eg (i) Convert into products of trig functions
) cos3 cos5a A A    
1 1
2sin 8 sin 2
2 2
A A  
 2sin 4 sinA A  
2sin 4 sinA A
) sin6 sin 4b x x  sin6 sin 4x x  
   
1 1
2sin 2 cos 10
2 2
x x
2sin cos5x x
(ii) Solve sin sin3 0 0 360x x x    
 2sin 2 cos 0x x 
2sin 2 cos 0x x 
sin 2 0 or cos 0x x 
2 0 ,180 ,360 ,540 ,720x      

42. eg (i) Convert into products of trig functions
) cos3 cos5a A A    
1 1
2sin 8 sin 2
2 2
A A  
 2sin 4 sinA A  
2sin 4 sinA A
) sin6 sin 4b x x  sin6 sin 4x x  
   
1 1
2sin 2 cos 10
2 2
x x
2sin cos5x x
(ii) Solve sin sin3 0 0 360x x x    
 2sin 2 cos 0x x 
2sin 2 cos 0x x 
sin 2 0 or cos 0x x 
2 0 ,180 ,360 ,540 ,720x      
0 ,90 ,180 ,270 ,360x      

43. eg (i) Convert into products of trig functions
) cos3 cos5a A A    
1 1
2sin 8 sin 2
2 2
A A  
 2sin 4 sinA A  
2sin 4 sinA A
) sin6 sin 4b x x  sin6 sin 4x x  
   
1 1
2sin 2 cos 10
2 2
x x
2sin cos5x x
(ii) Solve sin sin3 0 0 360x x x    
 2sin 2 cos 0x x 
2sin 2 cos 0x x 
sin 2 0 or cos 0x x 
2 0 ,180 ,360 ,540 ,720x      
0 ,90 ,180 ,270 ,360x      
90 ,270x   

44. eg (i) Convert into products of trig functions
) cos3 cos5a A A    
1 1
2sin 8 sin 2
2 2
A A  
 2sin 4 sinA A  
2sin 4 sinA A
) sin6 sin 4b x x  sin6 sin 4x x  
   
1 1
2sin 2 cos 10
2 2
x x
2sin cos5x x
(ii) Solve sin sin3 0 0 360x x x    
 2sin 2 cos 0x x 
2sin 2 cos 0x x 
sin 2 0 or cos 0x x 
2 0 ,180 ,360 ,540 ,720x      
0 ,90 ,180 ,270 ,360x      
90 ,270x   
0 ,90 ,180 ,270 ,360x      

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