The-equation-cosec-x-2-cosec-y-2-cosec-z-2-6-where-0-lt-x-y-z-lt-pi-2-and-x-y-z-pi-have-1-Three-ordered-triplet-x-y-z-solutions-2-Two-ordered-triplet-x-y-z-soluti

Question Number 18455 by Tinkutara last updated on 21/Jul/17

The equation cosec \frac{x}{2} + cosec \frac{y}{2} +

cosec \frac{z}{2} = 6, where 0 < x, y, z < \frac{π}{2} and

x + y + z = π, have

(1) Three ordered triplet (x, y, z)

solutions

(2) Two ordered triplet (x, y, z)

solutions

(3) Just one ordered triplet (x, y, z)

solution

(4) No ordered triplet (x, y, z) solution

Answered by Tinkutara last updated on 02/Aug/17

Commented by Tinkutara last updated on 02/Aug/17

Consider the graph of f (x) = cosec \frac{x}{2}

in [0, π] . Let A, B and C be three points

on this graph .

A = (x, cosec \frac{x}{2}) B = (y, cosec \frac{y}{2}),

C = (z, cosec \frac{z}{2})

Centroid of this triangle is

(\frac{x + y + z}{3}, \frac{cosec \frac{x}{2} + cosec \frac{y}{2} + cosec \frac{z}{2}}{3})

Corresponding to this x - coordinate,

the y - coordinate on the graph i . e .

f (\frac{x + y + z}{3}) = cosec \frac{x + y + z}{6} = 2

Clearly from the graph it can be seen

that \frac{cosec \frac{x}{2} + cosec \frac{y}{2} + cosec \frac{z}{2}}{3} ⩾ 2

\Rightarrow cosec \frac{x}{2} + cosec \frac{y}{2} + cosec \frac{z}{2} ⩾ 6

But given that cosec \frac{x}{2} + cosec \frac{y}{2} + cosec \frac{z}{2} = 6

Hence there is only one case of

x = y = z = \frac{π}{3} .