Pt usurinta calculelor, vom nota muchia cu a
Fie N, mijlocullui AC
atunci MN l.m. ΔDAC,MN||DC
cos∡(OM,DC)=cos∡(OM, MN)=cos∡(OMN)
MN=a/2
ON=1/3 inaltime tr echilat latura a =1/3 * a√3/2=a√3/6
OM =mediabna in tr dr de ipotenuza DA=a, OM=a/2
OMN tr isoscel baza ON, fie MP⊥ON, P∈ON
MP=√((a/2)²-(a√3/12)²)= a/2*√(1-(√3/6)²)=a/2 *√(1-1/12)=a/2*√11/√12
agalam aria OMN din 2 formule AN*MP/2=OM 8MN *sin∡ (OMN)/2
(a√3/6 )*a√11/2√12= a/2 *a/2 sin ∡(OMN)
efectuand calculele obtinem
sin∡ OMN=(√11)/6
aplicand formula fundamentala a trigonometriei, obtinem
cos ∡OMN=5/6
