Math Is Fun Forum

tony123

Member

Registered: 2007-08-03

Posts: 229

The value

cos(a-b) + cos(b-c) + cos(c-a) = -3/2

The value (sina + sinb + sinc)/(cosa + cosb + cosc) is:
(i)Some Rational Number
(ii)Infinity
(iii)0
(iv)Indeterminate

Avon

Member

Registered: 2007-06-28

Posts: 80

Re: The value

(sin(a) + sin(b) + sin(c))^2 + (cos(a) + cos(b) + cos(c))^2

= sin^2(a) + sin^2(b) + sin^2(c) + cos^2(a) + cos^2(b) + cos^2(c)
   + 2sin(a)sin(b) + 2sin(a)sin(c) + 2sin(b)sin(c)
   + 2cos(a)cos(b) + 2cos(a)cos(c) + 2cos(b)cos(c)

= 3 + 2(cos(a-b) + cos(b-c) + cos(c-a))
= 0

Assuming a,b,c are real we must have
sin(a) + sin(b) + sin(c) = cos(a) + cos(b) + cos(c) = 0
and so the answer is (iv).