Two sides of a triangle measure 8 and 15. How many integer values can be assigned to the third side so that all the angles in the triangle are acute?
The cosine rule gives the formula for the cosine of an angle when the three sides are known:
cos(A)=(B²+C²-A²)/(2BC)
where B and C are lengths of sides adjacent to A.
We know that a positive angle is acute when cos(A) is positive, which in turn requires that
B²+C²-A²>0 ....(1)
So if all three angles of a triangle have to be acute, we have the two additional requirement analogous to (1) above:
C²+A²-B²>0
A²+B²-C²>0
If the third side can be chosen such that all three conditions are satisfied, the triangle has acute angles.