Converting Sum or Difference into Product

We will learn how to deal with the formula for converting sum or difference into product.

(i) the sum of two sines into a product of a pair of sine and cosine

(ii) the difference of two sines into a product of a pair of cosine and sine

(iii) the sum of two cosines into a product of two cosines       

(iv) the difference of two cosines into a product of two sines

If X and Y are any two real numbers or angles, then

(a) sin (X + Y) + sin (X - Y) = 2 sin X cos Y

(b) sin (X + Y) - sin (X - Y) = 2 cos X sin Y

(c) cos (X + Y) + cos (X - Y) = 2 cos X cos Y

(d) cos (X - Y) - cos (X + Y) = 2 sin X sin Y

(a), (b), (c) and (d) are considered as formulae of transformation from sum or difference to product.

Proof:

(a) We know that sin (X + Y) = sin X cos Y + cos X sin Y ……… (i)   

and sin (X - Y) = sin X cos Y - cos X sin Y ……… (ii)               

Adding (i) and (ii) we get,

sin (X + Y) + sin (X - Y) = 2 sin X cos Y  ………………..… (1)  


(b) We know that sin (X + Y) = sin X cos Y + cos X sin Y ……… (i)  

and sin (X - Y) = sin X cos Y - cos X sin Y ……… (ii)               

Subtracting (ii) from (i) we get,

sin (X + Y) - sin (X - Y) = 2 cos X sin Y  ………………..… (2)   


(c) We know that cos (X + Y) = cos X cos Y + sin X sin Y ……… (iii)

and cos (X - Y) = cos X cos Y - sin X sin Y ……… (iv)                            

Adding (iii) and (iv) we get,

cos (X + Y) + cos (X - Y) = 2 cos X cos Y  ………………..… (3)


(d) We know that cos (X + Y) = cos X cos Y + sin X sin Y ……… (iii)               

and cos (X - Y) = cos X cos Y - sin X sin Y ……… (iv)                            

Subtracting (iii) from (iv) we get,

cos (X - Y) - cos (X + Y) = 2 sin X sin Y  ………………..… (4)  

Let, X + Y = α and X - Y = β.

Then, we have, X = (α + β)/2 and B = (α - β)/2.

Clearly, formula (1), (2), (3) and (4) reduce to the following forms in terms of C and D:

sin α + sin β = 2 sin (α + β)/2 cos (α - β)/2      ………. (5)

sin α - sin β = 2 cos (α + β)/2 sin (α - β)/2  ………  (6)

cos α + cos β = 2 cos (α + β)/2 cos (α - β)/2      ……… (7)

And cos α - cos β = -2 sin (α + β)/2 sin (α - β)/2

⇒ cos α - cos β = 2 sin (α + β)/2 sin (β - α)/2     ……… (8)

Note: (i) Formula sin α + sin β = 2 sin (α + β)/2 cos (α - β)/2 is transform the sum of two sines into a product of a pair of sine and cosine.

(ii) Formula sin α - sin β = 2 cos (α + β)/2 sin (α - β)/2 is transform the difference of two sines into a product of a pair of cosine and sine.

(iii) Formula cos α + cos β = 2 cos (α + β)/2 cos (α - β)/2 is transform the sum of two cosines into a product of two cosines.

(iv) Formula cos α - cos β = 2 sin (α + β)/2 sin (β - α)/2 is transforms the difference of two cosines into a product of two sines.






11 and 12 Grade Math

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