Question 1001510
In a right triangle the longest side is the hypotenuse,
and if {{{a}}}= length of the hypotenuse, with {{{b}}} and {{{c}}} being the other sides lengths,
{{{a^2=b^2+c^2}}} (the Pythagorean relationship).
In any triangle ABC,
{{{a^2=b^2+c^2-2bc*cos(A)}}} (the law of cosines).
(We can also write it as {{{b^2=a^2+c^2-2ac*cos(B)}}} or {{{c^2=a^2+b^2-2ab*cos(C)}}} ).
So if {{{a=16}}}= length of side opposite angle A
{{{16^2=13^3+11^2-2*13*11*cos(A)}}} 
{{{256=169+121-286*cos(A)}}} 
{{{286*cos(A)=169+121-256}}}
{{{286*cos(A)=34}}}
{{{cos(A)=34/286}}}--->{{{A=about1.4516}}} or {{{A=about83.17^o}}} (rounded)
NOTE:
We can also write the law of cosines as
{{{b^2=a^2+c^2-2ac*cos(B)}}} or {{{c^2=a^2+b^2-2ab*cos(C)}}} .


To find the measure of another angle,
we can apply the law of cosines again,
or we can use the law of sines.
After that, we find the remaining angle by using the fact that the measures of all 3 angles add up to {{{pi}}} or {{{180^o}}} .


USING LAW OF COSINES AGAIN:
let {{{B}}} be the angle opposite the side with length {{{b=13}}}cm.
{{{13^2=16^2+11^2-2*16*11*cos(B)}}}
{{{169=256+121-352*cos(B)}}}
{{{352*cos(B)=256+121-169}}}
{{{352*cos(B)=208}}}
{{{cos(B)=208/352}}}--->{{{B=about0.9386}}} or {{{B=about53.78^o}}} (rounded).


USING LAW OF SINES FOR A CHANGE:
If you have the measures of one side and the opposite angle of a triangle,
you can apply the law of sines, which states
{{{sin(B)/b=sin(A)/a=sin(C)/c}}}
So, with {{{b=13}}}cm,
{{{sin(B)/13=sin(A)/16}}}--->{{{sin(B)=13sin(A)/16}}}
{{{sin(A)=about0.9929}}} (rounded), so
{{{sin(B)=about13*0.9929/16}}}--->{{{B=about0.9386}}} or {{{B=about53.78^o}}} (rounded).


Once we have {{{system(A=about1.4516,B=about0.9386)}}} or {{{system(A=about83.17^o,B=about53.78^o)}}}
We can find
{{{C=pi-about1.4516-about0.9386=about0.7514}}} or
{{{C=180^o-83.17^o-about53.78^o=about43.05^o}}} .