F.4 Amaths

Sine rule:

(a / sin A) = (b / sin B) = (c / sin C) = k

i.e.　a = k sin A

　　b = k sin B

　　c = k sin C

(a)

　(a + b) / (a + b + c)

= (k sin A + k sin B) / (k sin A + k sin B + k sin C)

= (sin A + sin B) / (sin A + sin B + sin C)

(b)

　(a + b) / (a + b + c)

= (sin A + sin B) / (sin A + sin B + sin C)　, by (a)

= (sin A + sin 2A) / (sin A + sin 2A + sin (π - 3A))　, as B = 2A

= (sin A + sin 2A) / (sin A + sin 2A + sin 3A)

= (sin A + 2 sin A cos A) / (2 sin 2A cos A + sin 2A)

= [sin A (1 + 2 cosA)] / [sin 2A (1 + 2 cosA)]

= sin A / sin 2A

= sin A / sin B

= (k sin A) / (k sin B)

= a / b

(a)
Let a/sinA = b/sinB = c/sinC = k
Then, a = ksinA, b = ksinB, c = ksinC
L.H.S.
= (a + b) / (a + b + c)
= (ksinA + ksinB) / (ksinA + ksinB + ksinC)
= k(sinA + sinB) / k(sinA + sinB + sinC)
= (sinA + sinB) / (sinA + sinB + sinC)
= R.H.S.