Page 27 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes Similarly, by adding and subtracting the other two formulae, we get
2cos A cos B = cos(A + B) + cos(A B) …(3)
and 2sin A sin B = cos(A B) cos(A + B) …(4)
We can also reference these as
2sin A cos B =sin(sum) + sin(difference)
2cos A sin B =sin(sum) sin(difference)

2cos A cos B =cos (sum) + cos(difference)
2sin A sin B =cos(difference) cos(sum)

Transformation of sums or differences into products

In the above results put
A + B = C
A – B = D

Then A= C + D/2 and B = C – D/2 and (1), (2), (3) and (4) becomes
sin C + sin D = 2sin C + D/2 cos C – D/2
sin C – sin D = 2cos C + D/2 sin C – D/2
cos C + cos D = 2cos C + D/2 cos C – D/2
cos C – cos D = 2sin C + D/2 sin C – D/2

Further applications of addition and subtraction formulae

We shall prove that
2
2
(i) sin(A + B) sin(A B) = sin A sin B
2
2
2
2
(ii) cos(A + B) cos(A B) = cos A sin B or cos B sin A
Proof: (i) sin(A + B) sin(A B)
= ( sin A cos B + cos A sin B) (sin A cos B cos A sin B)
2
2
2
2
= sin A cos B cos A sin B
2
2
= sin A(1 sin B) (1 sin A) sin B
2
2
2
2
= sin A sin B
(ii) cos(A + B) cos(A B)
= (cos A cos B sin A sin B) (cos A cos B + sin A sin B)
= cos A cos B sin A sin B
2
2
2
2
2
2
2
2
= cos A(1 sin B) (1 cos A) sin B
= cos A sin B
2
2
= (1 sin A) (1 cos B)
2
2
2
2
= cos B sin A
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