Formulae for Converting Sum or Difference into Product

How to remember the formulae for converting sum or difference into product?

sin α + sin β = 2 sin (α + β)/2 cos (α - β)/2 ………. (i)

sin α - sin β = 2 cos (α + β)/2 sin (α - β)/2 ………. (ii)

cos α + cos β = 2 cos (α + β)/2 cos (α - β)/2 ………. (iii)

cos α - cos β = 2 sin (α + β)/2 sin (β - α)/2 ………. (iv)

The following points will help us to remember the above four formulas:

(i) In the product part 2 always appear as a factor.

(ii) The angles in sin or cos of product appear as sum/2 that is, (α + β)/2 of the given angles α and β.

(iii) The angles in sin or cos of product appear as difference/2 that is, (α - β)/2 of the given angles α and β.

(iv) But, there is an exception in the formula for cos α - cos β = 2 sin (α + β)/2 sin (β - α)/2, here we can see in place of (α - β)/2 we have (β - α)/2.

(v) In case of formula (i), the product consists of a pair of sin and cos in the conversion of the sum of two sines we get sin (α + β)/2 and cos (α - β)/2 as factors.

(vi) In case of formula (ii), the product consists of a pair of sin and cos in the conversion of the difference of two sines we get cos (α + β)/2 and sin (α - β)/2 as factors.

(vii) In case of formula (iii), the product consists of two cosines as factors in the conversion of the sum of two cosines.

(viii) In case of formula (iv), the product consists of two sines as factors in the conversion of difference of two cosines.

The following verbal statements will help us to remember the above four formulas:

For formula (i): sin + sin = 2 sin (sum/2) cos (difference/2)

For formula (ii): sin - sin = 2 cos (sum/2) sin (difference/2)

For formula (iii): cos + cos = 2 cos (sum/2) cos (difference/2)

For formula (iv): cos - cos = 2 sin (sum/2) sin (difference reversed/2)

11 and 12 Grade Math

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