I was trying to think of a set of five trig functions in which the first trig function is multiplied by some constant a, the second by a different constant b, the third by a different constant c, the fourth by a different constant d, and the fifth by a different constant f, such that if $a^2+b^2+c^2+d^2+f^2=g^2$, the trig functions when multiplied by their constants, squared, and then added together also produce $g^2$. The trig functions I thought might be solutions to this don't work and I haven't been able to find any that work.
What five trig functions would be solutions to this problem.
I think what you're really looking for is the hyperspherical coordinate system in 5 dimensions.
$$\begin{eqnarray}a & = & g\cos(\phi_1) \\ b & = & g\sin(\phi_1)\cos(\phi_2) \\ c & = & g\sin(\phi_1)\sin(\phi_2)\cos(\phi_3) \\ d & = & g\sin(\phi_1)\sin(\phi_2)\sin(\phi_3)\cos(\phi_4) \\ f & = & g\sin(\phi_1)\sin(\phi_2)\sin(\phi_3)\sin(\phi_4) \\ \end{eqnarray}$$
with $\forall i \in {1,2,3}:\phi_i \in [0,\pi]$ and $\phi_4 \in [0,2\pi]$. Then, $a^2+b^2+c^2+d^2+f^2=g^2$.