If-xcos-ycos-2pi-3-zcos-4pi-3-then-the-value-of-xy-yz-zx-

Question Number 27840 by bmind4860 last updated on 15/Jan/18

I f x c o s θ = y c o s (θ + \frac{2 π}{3}) = z c o s (θ + \frac{4 π}{3}), t h e n

t h e v a l u e o f x y + y z + z x .

Answered by ajfour last updated on 15/Jan/18

l e t x \cos θ = y \cos (θ + \frac{2 π}{3})

= z \cos (θ + \frac{4 π}{3}) = r

x y + y z + z x = x y z (\frac{1}{x} + \frac{1}{y} + \frac{1}{z})

= \frac{r^{3}}{\cos θ \cos (θ + \frac{2 π}{3}) \cos (θ + \frac{4 π}{3})} [\frac{\cos θ}{r} + \frac{\cos (θ + \frac{2 π}{3})}{r} + \frac{\cos (θ + \frac{4 π}{3})}{r}]

a n d s i n c e

\cos θ + \cos (θ + \frac{4 π}{3}) + \cos (θ + \frac{2 π}{3})

= 2 \cos (θ + \frac{2 π}{3}) \cos (\frac{2 π}{3}) + \cos (θ + \frac{2 π}{3})

= - \cos (θ + \frac{2 π}{3}) + \cos (θ + \frac{2 π}{3}) = 0

s o x y + y z + z x = 0 .

Commented by bmind4860 last updated on 15/Jan/18

s u p e r b! s i r