Given the sides 11,13,19, to calculate for the smallest angle using cosine rule, i.e;
a^2=b^2+c^2-2bc cos (A)
we shall proceed as follows;
The smallest angle is always opposite to the shortest side; Since 11 is our shortest side, the smallest angle is the angle opposite to this side. Thus;
let a=11 and our smallest angle be A
thus
11^2=13^2+19^2-2*13*19 Cos A
121=169+361-494 Cos A
121=530-494 Cos A
-409=-494 Cos A
dividing through by -494 we get;
Cos A=0.8279
thus
A=Cos(-1) 0.8279
A=34.113°≈34°
The answer is A=34°
