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) cos3θ cos5θ

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
= ( 2cos3θ cos5θ )
b) cos3θ cos5θ
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
= ( 2cos3θ cos5θ )
b) cos3θ cos5θ
2
1
= ( cos ( 3θ + 5θ ) + cos ( 3θ − 5θ ) )
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
= ( 2cos3θ cos5θ )
b) cos3θ cos5θ
2
1
= ( cos ( 3θ + 5θ ) + cos ( 3θ − 5θ ) )
2
1
= ( cos8θ + cos ( −2θ ) )
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
= ( 2cos3θ cos5θ )
b) cos3θ cos5θ
2
1
= ( cos ( 3θ + 5θ ) + cos ( 3θ − 5θ ) )
2
1
1
= ( cos8θ + cos 2θ )
= ( cos8θ + cos ( −2θ ) )
2
2

16. (ii) Evaluate 2sin 45o cos15o

17. (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 )

18. (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 )
= sin 60o + sin 30o

19. (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 )
= sin 60o + sin 30o
31
= +
22

20. (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 )
= sin 60o + sin 30o
31
= +
22
3 +1
=
2

21. (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 )
= sin 60o + sin 30o
31
= +
22
3 +1
=
2
Sums to Products

22. (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 )
= sin 60o + sin 30o
31
= +
22
3 +1
=
2
Sums to Products
1 1
sin A + sin B = 2sin ( A + B ) cos ( A − B )
2 2

23. (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 )
= sin 60o + sin 30o
31
= +
22
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 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 )
= sin 60o + sin 30o
31
= +
22
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 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 )
= sin 60o + sin 30o
31
= +
22
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 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 )
= sin 60o + sin 30o
31
= +
22
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 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 )
= sin 60o + sin 30o
31
= +
22
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 0o ≤ x ≤ 360o

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 0o ≤ x ≤ 360o
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 0o ≤ x ≤ 360o
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 0o ≤ x ≤ 360o
2sin 2 x cos ( − x ) = 0
2sin 2 x cos x = 0
sin 2 x = 0 cos x = 0
or

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 0o ≤ x ≤ 360o
2sin 2 x cos ( − x ) = 0
2sin 2 x cos x = 0
sin 2 x = 0 cos x = 0
or
2 x = 0o ,180o ,360o ,540o ,720o

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 0o ≤ x ≤ 360o
2sin 2 x cos ( − x ) = 0
2sin 2 x cos x = 0
sin 2 x = 0 cos x = 0
or
2 x = 0o ,180o ,360o ,540o ,720o
x = 0o ,90o ,180o , 270o ,360o

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 0o ≤ x ≤ 360o
2sin 2 x cos ( − x ) = 0
2sin 2 x cos x = 0
sin 2 x = 0 cos x = 0
or
x = 90o , 270o
2 x = 0o ,180o ,360o ,540o ,720o
x = 0o ,90o ,180o , 270o ,360o

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 0o ≤ x ≤ 360o
2sin 2 x cos ( − x ) = 0
2sin 2 x cos x = 0
sin 2 x = 0 cos x = 0
or
x = 90o , 270o
2 x = 0o ,180o ,360o ,540o ,720o
x = 0o ,90o ,180o , 270o ,360o ∴ x = 0o ,90o ,180o , 270o ,360o

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