Problem:
Pyramid OABCD has square base ABCD, congruent edges OA,OB,OC, and OD, and β AOB=45β. Let ΞΈ be the measure of the dihedral angle formed by faces OAB and OBC. Given that cosΞΈ=m+nβ, where m and n are integers, find m+n.
Solution:
Let P be the foot of the perpendicular from A to OB. The pyramid's symmetry implies that P is also the foot of the perpendicular from C to OB. Without loss of generality, we may assume OP=1, from which AP=PC=1,OB=OA=2β, and BP=2ββ1 follow. Two applications of the Pythagorean Theorem now give
AB2=AP2+BP2=4β22β
and
AC2=2β AB2=8β42β
The measure of the dihedral angle determined by faces OAB and OBC is the same as that of β APC. Use the Law of Cosines to obtain
