to ishuda
I tried that got nowhere
I don't know what answer to expect
angle a= 22.5 deg.
b =42.83
c =114.76
seems to work
Just a trial post with nothing else said.
to ishuda
I tried that got nowhere
I don't know what answer to expect
angle a= 22.5 deg.
b =42.83
c =114.76
seems to work
Not quite. cos(c) is negative, tan(a) is positive. I believe there is no solution. Since these equations are circular, i.e. replace a with b, b with c, and c with a and you have the same equations, the following applies for all three variables:
\(\displaystyle cos^2(a) + 1 = tan^2(b) + 1 = \dfrac{1}{cos^2(b)} = \dfrac{1}{tan^2(c) + 1 - 1}\)
\(\displaystyle =\dfrac{1}{\frac{1}{cos^2(c)} - 1} = \dfrac{cos^2(c)}{1 - cos^2(c)}=\dfrac{tan^2(a)}{1-tan^2(a)}\)
\(\displaystyle =\dfrac{sin^2(a)}{cos^2(a)-sin^2(a)}=\dfrac{1 - cos^2(a)}{2 cos^2(a) - 1}\)
or, simplifying,
\(\displaystyle 2 cos^4(a) + 2 cos^2(a) - cos^2(a) - 1 = 1 - cos^2(a)\)
or
\(\displaystyle cos^4(a) + cos^2(a) - 1 = 0\)
which gives
\(\displaystyle cos^2(a) = \dfrac{-1 \pm \sqrt{5}}{2}\)
since the minus sign gives complex roots which, for this problem, are not allowed we have
cos(a)=\(\displaystyle \pm \sqrt{0.5 (\sqrt{5} - 1)}\)
and a (and b and c) must be given by
a ~ 2.475353 [~141.827 deg] or a ~ 0.666239 [~38.173 deg]
plus or minus, of course, some number of complete cycles, i.e \(\displaystyle \pm 2 n \pi\). I don't think you can make a combination of those angles, all positive and less than 180 degrees individually, to add to \(\displaystyle \pi\)(180 degrees).
