Transformation of Trigonometric Formulae

We can transform the product of the trigonometric ratios of the angles into the sum or the difference of the trigonometric ratios of the compound angles and vice versa.

Transformation of products into sum or difference

(a) We know that,

sinA. cosB + cosA sinB = sin(A + B) .................(i)

sinA cosB - cosA sinB = sin(A - B) ...................(ii)

Adding (i) and (ii), we get

2sinA cosB = sin(A + B) + sin(A - B)

Subtracting (ii) from (i) we get.

2cosA sinB = sin(A + B) - sin(A - B)

(b) Again we know that,

cosA cosB - sinA sinB = cos(A + B) .............(iii)

cosA cosB + sinA sinB = cos(A - B) ..............(iv)

Adding (iii) and (iv) we get

2cosA cosB = cos(A + B) + cos(A - B)

Subtracting (iii) from (iv) we get

2sinA sinB = cos(A - B) - cos(A + B)

Now, the following formulae transform the product of the trigonometric ratios of the angles into the sum or the difference of the trigonometric ratios of the compound angles:

2 sinA cosB = sin(A + B) - sin(A - B)

2 cosA sinB = sin(A + B) - sin(A - B)

2 cosA cosB = sin(A + B) = cos(A - B)

2 sinA sinB = cos(A - B) - cos(A + B)

It will be convenient to remmember the above formulae in the form

2 sin cos = sin + sin

2 sin cos = sin - sin

2 cos cos = cos + cos

2 sin sin = cos - cos

Transformation of sum or difference into product

From the above formula, we have

sin(A + B) + sin(A - B) = 2 sinA cosB ...........(i)

sin(A + B) - sin(A - B) = 2 cosA sinB ........... (ii)

cos(A + B) + cos (A - B) = 2 cosA cosB ...............(iii)

cos(A - B) - cos(A + B) = 2 sinA sinB ..............(iv)

Suppose A + B = C and A - B = D

Adding the two we get, 2A = C + D

or, A =\(\frac{C + D}{2}\)

Again subtracting second from first, we get 2B = C - D

or, B =\(\frac{C - D}{2}\)

Now, substituting the values of A, B, A + B and A - B in (i), (ii), (iii) and (iv), we have

sinC + sinD = 2 sin (\()\frac{C + D}{2}\) cos (\(\frac{C - D}{2}\))

sinC - sinD = 2 cos (\(\frac{C + D}{2}\)) sin (\(\frac{C - D}{2}\))

cosC + cosD = 2 cos (\(\frac{C + D}{2}\)) cos (\(\frac{C - D}{2}\))

cosD - cosC = 2 sin (\(\frac{C + D}{2}\)) sin (\(\frac{D - C}{2}\))

cosC - cosD = -2 sin (\(\frac{C + D}{2}\)) sin (\(\frac{C - D}{2}\))

These formulae transform the sum or difference of trigonometric ratios into the products of trigonometric ratios.

It will be convenient to remember the above formulae in the form

sin + sin = 2 sin. cos

sin - sin = 2 cos. sin

cos + cos = 2 cos. sin

cos - cos = 2 sin. sin