If side $a$ is known and the angles are given as functions of two variables (let's call them $x$ and $y$), what is the easiest way to find $y$ as a function of $x$. To make things easier, let one of the angles be $y$.
I'm actually stuck with a concrete problem. I've tried using and combining different trigonometric identities, but all the efforts were to no avail.
EDIT: So, to write problem as it is: Angles are $y$, $\frac \pi2 +x-y$, and $\frac \pi 2 -x$. $a$ is the side that lies opposite to angle $\frac \pi2 +x-y$.
What makes it possible to solve equations concerning the angles of a triangle is that the three angles add up to a constant $\pi$, or $180$ degrees.
Now, if you have expressions in terms of $x$ and $y$ for two of the angles but not the third, $A=y$, $B=f(x,y)$, and $C=180-A-B$, then $C$ destroys the usefulness of the context of the triangle, absorbing all of the problem solving potential.
This is because you need one equation for each unknown if you are going to find a specific value for everything, or minus one for each value you plug into the function if you're looking for a function. To get $y$ as a function of $x$, you need one equation, plus another equation each for any other unknowns added. You have two unknowns, $x$ and $y$, and the definition of $C$ introduces $C$ as a third unknown in the middle of the only equation you had.
So, you need $C$ expressed by a third expression that does not use the expressions for $A$ and $B$. I think it's okay if it uses one or the other, but you need to be able to equate two different expressions at the end in order to have any meaning. It must not reduce to $180$ degrees $=180$ degrees when you plug the expressions for the angles into $A+B+C=180$ degrees. If you have $A=y$, $B=f(x,y)$ and $C=g(x,y)$, then you can plug them all into $A+B+C=\pi$ and isolate $y$ to get $y$ as a function of $x$.
Does this solve your problem?