Transcript
Page 1: 11X1 T07 08 products to sums etc (2011)

Products to Sums

Page 2: 11X1 T07 08 products to sums etc (2011)

Products to Sums 2sin cos sin sinA B A B A B

Page 3: 11X1 T07 08 products to sums etc (2011)

Products to Sums 2sin cos sin sinA B A B A B

2cos cos cos cosA B A B A B

Page 4: 11X1 T07 08 products to sums etc (2011)

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

Page 5: 11X1 T07 08 products to sums etc (2011)

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

Page 6: 11X1 T07 08 products to sums etc (2011)

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

Page 7: 11X1 T07 08 products to sums etc (2011)

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

Page 8: 11X1 T07 08 products to sums etc (2011)

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

Page 9: 11X1 T07 08 products to sums etc (2011)

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

sin 6 sin 4x x

Page 10: 11X1 T07 08 products to sums etc (2011)

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

sin 6 sin 4x x

sin 6 sin 4x x

Page 11: 11X1 T07 08 products to sums etc (2011)

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

sin 6 sin 4x x

sin 6 sin 4x x

) cos3 cos5b

Page 12: 11X1 T07 08 products to sums etc (2011)

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

sin 6 sin 4x x

sin 6 sin 4x x

) cos3 cos5b 1 2cos3 cos52

Page 13: 11X1 T07 08 products to sums etc (2011)

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

sin 6 sin 4x x

sin 6 sin 4x x

) cos3 cos5b

1 cos 3 5 cos 3 52

1 2cos3 cos52

Page 14: 11X1 T07 08 products to sums etc (2011)

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

sin 6 sin 4x x

sin 6 sin 4x x

) cos3 cos5b

1 cos 3 5 cos 3 52

1 cos8 cos 22

1 2cos3 cos52

Page 15: 11X1 T07 08 products to sums etc (2011)

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

sin 6 sin 4x x

sin 6 sin 4x x

) cos3 cos5b

1 cos 3 5 cos 3 52

1 cos8 cos 22

1 cos8 cos 22

1 2cos3 cos52

Page 16: 11X1 T07 08 products to sums etc (2011)

(ii) Evaluate 2sin 45 cos15

Page 17: 11X1 T07 08 products to sums etc (2011)

(ii) Evaluate 2sin 45 cos15 sin 45 15 sin 45 15

Page 18: 11X1 T07 08 products to sums etc (2011)

(ii) Evaluate 2sin 45 cos15 sin 45 15 sin 45 15

sin 60 sin30

Page 19: 11X1 T07 08 products to sums etc (2011)

(ii) Evaluate 2sin 45 cos15 sin 45 15 sin 45 15

sin 60 sin30

3 12 2

Page 20: 11X1 T07 08 products to sums etc (2011)

(ii) Evaluate 2sin 45 cos15 sin 45 15 sin 45 15

sin 60 sin30

3 12 2

3 12

Page 21: 11X1 T07 08 products to sums etc (2011)

(ii) Evaluate 2sin 45 cos15 sin 45 15 sin 45 15

sin 60 sin30

3 12 2

3 12

Sums to Products

Page 22: 11X1 T07 08 products to sums etc (2011)

(ii) Evaluate 2sin 45 cos15 sin 45 15 sin 45 15

sin 60 sin30

3 12 2

3 12

Sums to Products 1 1sin sin 2sin cos

2 2A B A B A B

Page 23: 11X1 T07 08 products to sums etc (2011)

(ii) Evaluate 2sin 45 cos15 sin 45 15 sin 45 15

sin 60 sin30

3 12 2

3 12

Sums to Products 1 1sin sin 2sin cos

2 2A B A B A B (2 sine half sum cos half diff)

Page 24: 11X1 T07 08 products to sums etc (2011)

(ii) Evaluate 2sin 45 cos15 sin 45 15 sin 45 15

sin 60 sin30

3 12 2

3 12

Sums to Products 1 1sin sin 2sin cos

2 2A B A B A B (2 sine half sum cos half diff)

1 1cos cos 2cos cos2 2

A B A B A B

Page 25: 11X1 T07 08 products to sums etc (2011)

(ii) Evaluate 2sin 45 cos15 sin 45 15 sin 45 15

sin 60 sin30

3 12 2

3 12

Sums to Products 1 1sin sin 2sin cos

2 2A B A B A B (2 sine half sum cos half diff)

1 1cos cos 2cos cos2 2

A B A B A B (2 cos half sum cos half diff)

Page 26: 11X1 T07 08 products to sums etc (2011)

(ii) Evaluate 2sin 45 cos15 sin 45 15 sin 45 15

sin 60 sin30

3 12 2

3 12

Sums to Products 1 1sin sin 2sin cos

2 2A B A B A B (2 sine half sum cos half diff)

1 1cos cos 2cos cos2 2

A B A B A B (2 cos half sum cos half diff)

1 1cos cos 2sin sin2 2

A B A B A B

Page 27: 11X1 T07 08 products to sums etc (2011)

(ii) Evaluate 2sin 45 cos15 sin 45 15 sin 45 15

sin 60 sin30

3 12 2

3 12

Sums to Products 1 1sin sin 2sin cos

2 2A B A B A B (2 sine half sum cos half diff)

1 1cos cos 2cos cos2 2

A B A B A B (2 cos half sum cos half diff)

1 1cos cos 2sin sin2 2

A B A B A B (minus 2 sine half sum sine half diff)

Page 28: 11X1 T07 08 products to sums etc (2011)

eg (i) Convert into products of trig functions

Page 29: 11X1 T07 08 products to sums etc (2011)

eg (i) Convert into products of trig functions

) cos3 cos5a A A

Page 30: 11X1 T07 08 products to sums etc (2011)

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 22 2

A A

Page 31: 11X1 T07 08 products to sums etc (2011)

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 22 2

A A

2sin 4 sinA A

Page 32: 11X1 T07 08 products to sums etc (2011)

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 22 2

A A

2sin 4 sinA A

2sin 4 sinA A

Page 33: 11X1 T07 08 products to sums etc (2011)

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 22 2

A A

2sin 4 sinA A

2sin 4 sinA A

) sin 6 sin 4b x x

Page 34: 11X1 T07 08 products to sums etc (2011)

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 22 2

A A

2sin 4 sinA A

2sin 4 sinA A

) sin 6 sin 4b x x sin 6 sin 4x x

Page 35: 11X1 T07 08 products to sums etc (2011)

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 22 2

A A

2sin 4 sinA A

2sin 4 sinA A

) sin 6 sin 4b x x sin 6 sin 4x x

1 12sin 2 cos 102 2

x x

Page 36: 11X1 T07 08 products to sums etc (2011)

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 22 2

A A

2sin 4 sinA A

2sin 4 sinA A

) sin 6 sin 4b x x sin 6 sin 4x x

1 12sin 2 cos 102 2

x x

2sin cos5x x

Page 37: 11X1 T07 08 products to sums etc (2011)

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 22 2

A A

2sin 4 sinA A

2sin 4 sinA A

) sin 6 sin 4b x x sin 6 sin 4x x

1 12sin 2 cos 102 2

x x

2sin cos5x x(ii) Solve sin sin 3 0 0 360x x x

Page 38: 11X1 T07 08 products to sums etc (2011)

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 22 2

A A

2sin 4 sinA A

2sin 4 sinA A

) sin 6 sin 4b x x sin 6 sin 4x x

1 12sin 2 cos 102 2

x x

2sin cos5x x(ii) Solve sin sin 3 0 0 360x x x

2sin 2 cos 0x x

Page 39: 11X1 T07 08 products to sums etc (2011)

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 22 2

A A

2sin 4 sinA A

2sin 4 sinA A

) sin 6 sin 4b x x sin 6 sin 4x x

1 12sin 2 cos 102 2

x x

2sin cos5x x(ii) Solve sin sin 3 0 0 360x x x

2sin 2 cos 0x x 2sin 2 cos 0x x

Page 40: 11X1 T07 08 products to sums etc (2011)

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 22 2

A A

2sin 4 sinA A

2sin 4 sinA A

) sin 6 sin 4b x x sin 6 sin 4x x

1 12sin 2 cos 102 2

x x

2sin cos5x x(ii) Solve sin sin 3 0 0 360x x x

2sin 2 cos 0x x 2sin 2 cos 0x x

sin 2 0 or cos 0x x

Page 41: 11X1 T07 08 products to sums etc (2011)

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 22 2

A A

2sin 4 sinA A

2sin 4 sinA A

) sin 6 sin 4b x x sin 6 sin 4x x

1 12sin 2 cos 102 2

x x

2sin cos5x x(ii) Solve sin sin 3 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

Page 42: 11X1 T07 08 products to sums etc (2011)

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 22 2

A A

2sin 4 sinA A

2sin 4 sinA A

) sin 6 sin 4b x x sin 6 sin 4x x

1 12sin 2 cos 102 2

x x

2sin cos5x x(ii) Solve sin sin 3 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

Page 43: 11X1 T07 08 products to sums etc (2011)

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 22 2

A A

2sin 4 sinA A

2sin 4 sinA A

) sin 6 sin 4b x x sin 6 sin 4x x

1 12sin 2 cos 102 2

x x

2sin cos5x x(ii) Solve sin sin 3 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

Page 44: 11X1 T07 08 products to sums etc (2011)

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 22 2

A A

2sin 4 sinA A

2sin 4 sinA A

) sin 6 sin 4b x x sin 6 sin 4x x

1 12sin 2 cos 102 2

x x

2sin cos5x x(ii) Solve sin sin 3 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

Page 45: 11X1 T07 08 products to sums etc (2011)

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


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