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# 11 X1 T04 14 Products To Sums Etc

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### 11 X1 T04 14 Products To Sums Etc

1. 1. Products to Sums
2. 2. Products to Sums 2sin A cos B = sin ( A + B ) + sin ( A − B )
3. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 16. (ii) Evaluate 2sin 45o cos15o
17. 17. (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 )
18. 18. (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 ) = sin 60o + sin 30o
19. 19. (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 ) = sin 60o + sin 30o 31 = + 22
20. 20. (ii) Evaluate 2sin 45o cos15o = sin ( 45 + 15 ) + sin ( 45 − 15 ) = sin 60o + sin 30o 31 = + 22 3 +1 = 2
21. 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. 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. 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. 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. 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. 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. 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. 28. eg (i) Convert into products of trig functions
29. 29. eg (i) Convert into products of trig functions a ) cos3 A − cos5 A
30. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 45. Exercise 2F; 1b, 2b, 3a, 9, 10ace