Joshua Foley

2022-07-04

Consider the system of equations in real numbers satisfying

and . Find .
I substituted as , where

toriannucz

Expert

to any values $a,b,c$ instead of $\frac{1}{4},\frac{1}{5},\frac{1}{6}$
If we put

then LAcarguy’s method leads to the following explicit formulas for the solutions :

where $\epsilon$ is $+1$ or $-1$. In your initial question where $a=\frac{1}{4},b=\frac{1}{5},c=\frac{1}{6}$, one finds $G=\frac{7}{921600}$,

Ayaan Barr

Expert

Starting with $\frac{sint}{4}=\frac{sinu}{5}=\frac{sinw}{6}$, then use the law of sines to get: $\frac{a}{4}=\frac{b}{5}=\frac{c}{6}$. So $a=\frac{2c}{3}$, $b=\frac{5c}{6}$. So apply the law of cosines to have: ${a}^{2}={b}^{2}+{c}^{2}-2bccosA$. Substituting these values of $a$ and $b$ into the above equation we can solve for $cosA$, and then $tanB$, and $tanA$, and $tanC$.