• Derive Law of sines.
Consider a triangle ABC , in which a ,b, c are the measures of sides
opposite to the angle 𝛼 , 𝛽 & 𝛾 respectively as show in Figure. Take a
rectangular coordinate system , in order to place the point C in the
standard position. The coordinate of point A will be ൫𝑏 cos 𝛾 , 𝑏 sin 𝛾൯.
If the point B is taken as the origin and measure of the angle
XBA =൫1800 − 𝛽൯ then the point A will have the coordinate
൫𝑐 cos൫1800 − 𝛽൯, 𝑐 sin൫1800 − 𝛽൯൯.
Since y co-ordinate is same in both cases;
We have , 𝑏 sin 𝛾 = 𝑐 sin൫1800 − 𝛽൯
𝑏 sin 𝛾 = 𝑐 sin 𝛽
𝑏 𝑐
= ………..(i)
sin 𝛽 sin 𝛾
Similarly , we have
𝑎 𝑐
= ……….(ii)
sin 𝛼 sin 𝛾
From eqn (i) & (ii) , we get
𝑎 𝑏 𝑐
= =
sin 𝛼 sin 𝛽 sin 𝛾
Consider a triangle ABC , in which a ,b, c are the measures of sides
opposite to the angle 𝛼 , 𝛽 & 𝛾 respectively as show in Figure. Take a
rectangular coordinate system , in order to place the point C in the
standard position. The coordinate of point A will be ൫𝑏 cos 𝛾 , 𝑏 sin 𝛾൯.
If the point B is taken as the origin and measure of the angle
XBA =൫1800 − 𝛽൯ then the point A will have the coordinate
൫𝑐 cos൫1800 − 𝛽൯, 𝑐 sin൫1800 − 𝛽൯൯.
Since y co-ordinate is same in both cases;
We have , 𝑏 sin 𝛾 = 𝑐 sin൫1800 − 𝛽൯
𝑏 sin 𝛾 = 𝑐 sin 𝛽
𝑏 𝑐
= ………..(i)
sin 𝛽 sin 𝛾
Similarly , we have
𝑎 𝑐
= ……….(ii)
sin 𝛼 sin 𝛾
From eqn (i) & (ii) , we get
𝑎 𝑏 𝑐
= =
sin 𝛼 sin 𝛽 sin 𝛾
