Plane Trigonometry and Numerical Computation

30. Law of Sines. Consider any triangle ABC with the altitude CD drawn from the vertex C (Fig. 28).

Fig. 28

In all cases we have

         h            h
sin A = ———, sin B = ———.                         (1)
         b            a

 sin A     a          a         b
——————— = ———,  or ——————— = ———————.             (2)
 sin B     b        sin B     sin B

If the perpendicular were dropped from B, the same argument would give a/sin A = c/sin C. Hence, we have

   a         b         c
——————— = ——————— = ———————. 
 sin A     sin B     sin C

This law is known as the law of sines and may be stated as follows: Any two sides of a triangle are proportional to the sines of the angles opposite these sides.

31. Law of Cosines. Consider any triangle ABC with the altitude CD drawn from the vertex C (Fig. 29).

In Fig. 29 a

AD =  b cos A; CD = b sin A; DB = c - b cos A. 

In Fig. 29 b

AD = -b cos A; CD = b sin A; DB = c - b cos A. 

In both figures

a² = DB² + CD².