Trigonometry! #130

Geometry Level 2

If $x=y\cos\frac{2\pi}{3}=z\cos\frac{4\pi}{3}$ then find the value of $xy+yz+zx$

This problem is part of the set Trigonometry .

1 solution

Caeo Tan
Jun 19, 2015

since cos(2pi/3)=cos(4pi/3)=-1/2

Eq. 1: x=y(-1/2)=z(-1/2)

xy+yz+xz=?

Method 1

from equation 1 we can get the values for y and z y=-2x

z=-2x

x(-2x)+(-2x)(-2x)+x(-2x)

-2x^2+4x^2+(-2x^2)

simplifying this we get 0

Method 2

xy+yz+xz=?

z=y and x=-y/2

(-y/2)(y)+y(y)+(-y/2)(y)=-(y^2)/2+y^2-(y^2)/2=0

Method 3

xy+yz+xz=?

x=-z/2 and y=z

(-z/2)(z)+z(z)+(-z/2)(z)=-(z^2)/2+z^2-(z^2)/2=0