Question

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?

Answer 1

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.