The equation I want to solve is:

\[ \tan \left( \tfrac{1}{2} C d^2 \right) - \tan \left( \tfrac{1}{2} C a^2 \right) = 0 \]

where \( \deta \) is given. I first plotted it to make sure it did in fact show trends of a root or roots for \( \delta = 0.5\).

delta:0.5; eqn(x):= tan(0.5*x*(delta^2)) - tan(0.5*x); plot2d(eqn, [x, 0, 20],[y, -1, 1]);

So, yup, it definitely has roots. The first root is actually at zero, but this is trivial in my case. So I am interesting in the second root. It seems to be around 8. I choose the interval for the root finder to look on as 7.8 to 8.5.

find_root(tan(0.5*x*(delta^2)) - tan(0.5*x), x, 7.8, 8.5);8.377580409572781

So the root found is the bold number above!

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