Law of Sines for Triangles on Plane Surface

1. The Law of Sines for Triangles on Plane Surface (also known as Euclidean Surface) states that for any triangle \(\vartriangle ABC\) (for e.g. the one given in the figure above) through 3 given points A, B and C having lengths of sides \(a\), \(b\) and \(c\) opposite to \(\angle A\), \(\angle B\) and \(\angle C\) respectively and \(i\), \(j\) and \(k\) the length of the perpendicular drawn from points A, B and C to sides with length \(a\),\(b\) and \(c\) respectively at points P, Q and R respectively, the following relation holds true
   
   \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\)
   
   \(\Rightarrow \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)
2. The following gives the derivation of the Law of Sines for Triangles on Plane Surface
   
   \(\sin A = \frac{j}{c}\) \(\Rightarrow c \sin A=j\)   ...(1)
   
   Also,
   
   \(\sin C = \frac{j}{a}\) \(\Rightarrow a \sin C=j\)   ...(2)
   
   From equation (1) and (2)
   
   \(c \sin A = a \sin C\)  \(\Rightarrow \frac{\sin A}{a}=\frac{\sin C}{c}\)    \(\Rightarrow \frac{a}{\sin A}=\frac{c}{\sin C}\)    ...(3)
   
   Now,
   
   \(\sin B = \frac{i}{c}\) \(\Rightarrow c \sin B=i\)   ...(4)
   
   Also,
   
   \(\sin C = \frac{i}{b}\) \(\Rightarrow b \sin C=i\)   ...(5)
   
   From equation (4) and (5)
   
   \(c \sin B = b \sin C\)  \(\Rightarrow \frac{\sin B}{b}=\frac{\sin C}{c}\)  \(\Rightarrow \frac{b}{\sin B}=\frac{c}{\sin C}\)   ...(6)
   
   From equation (3) and (6)
   
   \(\frac{\sin A}{a}=\frac{\sin B}{b}\)    \(\Rightarrow \frac{a}{\sin A}=\frac{b}{\sin B}\)
   
   \(\Rightarrow \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\)
   
   \(\Rightarrow \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)
3. The Law of Sines can be used to Find out the Remaining Sides and Angles of a Triangle when
   1. One Side and any Two Angles are given
   2. Two Sides and an Angle Not Between the Two given Sides are given