Page 25 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes Addition formulae
For any two numbers A and B,

cos (A + B) = cos A cosB sinA sinB

In given figure trace out
SOP = A
POQ = B

SOR = B
Where points P, Q, R, S lie on the unit circle.
Coordinates of P, Q, R, S will be (cos A, sin A),
[cos (A + B), sin (A + B)],
[cos ( B), sin ( B)], and (1, 0).

From the given figure, we have

side OP = side OQ
POR = QOS (each angle = B + QOR)
side OR = side OS

∆POR ∆QOS (by SAS)
PR = QS
2
PR = (cos A cos B) + (sin A sin( B) 2
QS = (cos A + B 1) + (sin A + B 0) 2
2
2
Since PR = QS 2
2
cos A + cos B 2cos A cos B + sin A + sin B + 2sin A sin B
2
2
2
= cos (A + B) + 1 2cos(A + B) + sin (A + B)
2
2
1 + 1 2(cos A cos B sin A sin B) = 1 + 1 2cos(A + B)
cosA cosB sinA sinB = cos (A + B) (I)
For any two numbers A and B, cos (A B) = cos A cos B + sin A sin B
Proof: Replace B by B in (I)
cos(A – B) = cos A cos B + sin A sin B
cos (–B) = cos B and sin(–B) = –sin B
For any two numbers A and B
sin(A + B) = sin A cos B + cos A sin B

Proof: We know that cos ( /2 – A) = sin A
sin ( /2 – A) = cos A
sin(A + B) = cos[ /2 – (A + B)
= cos[( /2 – A) + B]

= cos( /2 A) cos B + Sin ( /2 A)

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