If x = ycos(2pi/3) = zcos(4pi/3), then xy + yz + zx is equal to - Maths - Trigonometric Functions

Question

If x = ycos(2pi/3) = zcos(4pi/3), then xy + yz + zx is equal
to?????

Lalit Mehra · Accepted Answer

Dear Student!

Given, $x = y \cos (\frac{2 π}{3}) = z \cos (\frac{4 π}{3})$

$\cos (\frac{2 π}{3}) = \cos (π - \frac{π}{3}) = - \cos (\frac{π}{3}) = - \frac{1}{2} (∵ \cos (π - θ) = - \cos θ) \cos (\frac{4 π}{3}) = \cos (π + \frac{π}{3}) = - \cos (\frac{π}{3}) = - \frac{1}{2} (∵ \cos (π + θ) = - \cos θ)$

$x = y \cos (\frac{2 π}{3}) = z \cos (\frac{4 π}{3})$

$∴ x = - \frac{1}{2} y = - \frac{1}{2} z \Rightarrow x = - \frac{1}{2} y a n d x = - \frac{1}{2} z \Rightarrow y = - 2 x a n d z = - 2 x$

$x y + y z + z x = x (- 2 x) + (- 2 x) (- 2 x) + (- 2 x) x = - 2 x^{2} + 4 x^{2} - 2 x^{2} = 0$

Thus, the value of xy + yz + zx is 0.

Cheers!

If x = ycos(2pi/3) = zcos(4pi/3), then xy + yz + zx is equal to?????