11 x1 t04 14 products to sums etc

Products to Sums

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

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

2cos cos cos cosA B A B A B

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

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

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

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

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

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

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

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

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 12cos3 cos5

2

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

1cos 3 5 cos 3 5

2

12cos3 cos5

2

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

1cos 3 5 cos 3 5

2

1cos8 cos 2

2

12cos3 cos5

2

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

1cos 3 5 cos 3 5

2

1cos8 cos 2

2 1

cos8 cos 22

12cos3 cos5

2

(ii) Evaluate 2sin 45 cos15

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

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

sin 60 sin30

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

sin 60 sin30

3 12 2

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

sin 60 sin30

3 12 2

3 12

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

sin 60 sin30

3 12 2

3 12

Sums to Products

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

sin 60 sin30

3 12 2

3 12

Sums to Products 1 1

sin sin 2sin cos2 2

A B A B A B

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

sin 60 sin30

3 12 2

3 12

Sums to Products 1 1

sin sin 2sin cos2 2

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

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

sin 60 sin30

3 12 2

3 12

Sums to Products 1 1

sin sin 2sin cos2 2

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

1 1cos cos 2cos cos

2 2A B A B A B

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

sin 60 sin30

3 12 2

3 12

Sums to Products 1 1

sin sin 2sin cos2 2

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

1 1cos cos 2cos cos

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

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

sin 60 sin30

3 12 2

3 12

Sums to Products 1 1

sin sin 2sin cos2 2

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

1 1cos cos 2cos cos

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

1 1cos cos 2sin sin

2 2A B A B A B

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

sin 60 sin30

3 12 2

3 12

Sums to Products 1 1

sin sin 2sin cos2 2

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

1 1cos cos 2cos cos

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

1 1cos cos 2sin sin

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

sine half diff)

eg (i) Convert into products of trig functions

eg (i) Convert into products of trig functions

) cos3 cos5a A A

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 2

2 2A A

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 2

2 2A A

2sin 4 sinA A

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 2

2 2A A

2sin 4 sinA A

2sin 4 sinA A

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 2

2 2A A

2sin 4 sinA A

2sin 4 sinA A

) sin 6 sin 4b x x

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 2

2 2A A

2sin 4 sinA A

2sin 4 sinA A

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

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 2

2 2A A

2sin 4 sinA A

2sin 4 sinA A

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

1 12sin 2 cos 10

2 2x x

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 2

2 2A A

2sin 4 sinA A

2sin 4 sinA A

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

1 12sin 2 cos 10

2 2x x

2sin cos5x x

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 2

2 2A A

2sin 4 sinA A

2sin 4 sinA A

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

1 12sin 2 cos 10

2 2x x

2sin cos5x x(ii) Solve sin sin3 0 0 360x x x

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 2

2 2A A

2sin 4 sinA A

2sin 4 sinA A

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

1 12sin 2 cos 10

2 2x x

2sin cos5x x(ii) Solve sin sin3 0 0 360x x x

2sin 2 cos 0x x

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 2

2 2A A

2sin 4 sinA A

2sin 4 sinA A

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

1 12sin 2 cos 10

2 2x x

2sin cos5x x(ii) Solve sin sin3 0 0 360x x x

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

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 2

2 2A A

2sin 4 sinA A

2sin 4 sinA A

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

1 12sin 2 cos 10

2 2x 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

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 2

2 2A A

2sin 4 sinA A

2sin 4 sinA A

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

1 12sin 2 cos 10

2 2x 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

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 2

2 2A A

2sin 4 sinA A

2sin 4 sinA A

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

1 12sin 2 cos 10

2 2x 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

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 2

2 2A A

2sin 4 sinA A

2sin 4 sinA A

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

1 12sin 2 cos 10

2 2x 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

eg (i) Convert into products of trig functions

) cos3 cos5a A A 1 12sin 8 sin 2

2 2A A

2sin 4 sinA A

2sin 4 sinA A

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

1 12sin 2 cos 10

2 2x 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

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