suppose that we have Cartesian coordinate system.and suppose that we have three point which depend on parameter $t$,where t belongs to $(0,1)$;points are
$A(cos(3-t),sin(3-t))$
$B(cos(t),sin(t))$
$C(-cos(t),-sin(t))$
goal: find $t$ for which area of triangle $ABC$ is maximum
first of all,i was thinking that we could find length of each side of triangles,for example
$BC=2$
but what about another sides?we can use determinant formula like here
http://people.richland.edu/james/lecture/m116/matrices/applications.html
and goal will be find maximum determinant,but could we it?also i have calculate length of $AB$,which is equal $2*cos(t)*cos(3-t)-2*sin(t)*sin(3-t)$ which is i think
$2*cos(\alpha-\beta)$
or in our case it would be
$2*cos(t-(3-t))=2*cos(2*t-3)$ am on the right way?or could i simplify way of solution? EDITED: so rotation matrix in 2D has form
Hint: All these points lie on the unit circle. In particular, $B$ and $C$ are antipodal, meaning that the line segment $BC$ passes through the origin. Now, forget about point $A$ as in the problem. Where you should put a point $A'$ such that the area $A'BC$ is maximized, where $BC$ is an antipodal line segment? The solution of this problem is to make $A'BC$ a right triangle (You need to prove this). Now, see if your parametric equations form the same right triangle (up to rotations); and by the way, they will.