VITEEE 2014 Paper

Given: a = cos α + i sin α
b = cos β + i sin β
and c = cos γ + i sin γ
Now ,

b

c

=

cosβ+isinβ

cosγ+isinγ

×

cosγ−isinγ

cosγ−isinγ

cos β . cos γ + sin β . sin γ + i [ sin β cos γ - sin γ . cos β]
⇒

b

c

= cos (β - γ) + i sin (β - γ) ... (i)
Similarly,

x

a

= cos (γ - α) + i sin (β - γ) ... (ii)
and

a

b

= cos (α - β) + i sin (α - β) ... (iii)
On adding Eqs. (i), (ii) and (iii), we get
cos (β - α) + cos (γ - α) + cos (α + β) + i
[sin (β - γ) + sin (γ - α) + sin (α - β)] = 1
[Since b/a + c/a + a/b = 1]
On equating real parts we get
cos (β - γ) + cos (γ - α) + sin (α - β) = 1