Trigo=Algebra

Algebra Level 3

If $x$ and $y$ are real numbers and $z$ lies in the interval $[0,2π]$ , solve the following equation:

$\sin^5 z = 2x^2+5y^2+6xy-26x-42y+90$

Enter your answer as $x+y+\frac{6z}{π}$ .

1 solution

A Former Brilliant Member
Nov 13, 2017

The given equation can be rewritten as :

$\sin^5z= (x^2+4y^2+64+4xy-32y-16x)+(x^2+y^2+25+2xy-10x-10y)+1$

$\sin^5z= (x+2y-8)^2+(x+y-5)^2+1$

In above eqn : $max(LHS)=min(RHS)$

Hence we have :

$x+2y=8$

$x+y=5$ and

$z=\frac{π}{2}$

Solving the equations in $x$ and $y$ we get : $x=2,y=3$