Trigonometry (11th Edition) Clone

The law of sines for a triangle with side $a,b,c$ and angle $A, B, C$ is $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$ Now if we are given the lengths of 3 sides $a,b$ and $c$, the task is to find the values of angles $A, B$ and $C$. However, it is impossible to find them since no matter how we apply the law of sines, there are always 2 unknown variables, which means they cannot be solved with only one equation. For example, from the law of sines, we can deduce that $$\frac{a}{\sin A}=\frac{b}{\sin B}$$ Here $a$ and $b$ are known variables, but both $A$ and $B$ are unknown variables. With just this equation, we cannot calculate either $A$ or $B$ as there are 2 unknown variables here. Another example with sides $b$, $c$ and angles $B$, $C$ $$\frac{b}{\sin B}=\frac{c}{\sin C}$$ Again, only $b$ and $c$ are known, and $B$ and $C$ are unknown. With just this equation, $B$ and $C$ cannot be calculated. There must be 3 known variables and only one unknown variable. Therefore, given only the lengths of 3 sides, we cannot apply the law of sines to solve the triangle because we will also end up with an equation with 2 unknown variables, and it is not possible to solve 2 unknown variables with just one equation.