OD

OA

OB

OC

\alpha

\beta

\gamma

\gamma

\alpha=45^{\circ}

\beta=60^{\circ}

60^{\circ}

OA=a

OB=b

OAD

OBD

OD=\frac{OA}{\cos\alpha}=\frac{OA}{\cos45^{\circ}}=a\sqrt{2},

OD=\frac{OB}{\cos\beta}=\frac{b}{\cos60^{\circ}}=2b.

b=\frac{1}{2}OD=\frac{a\sqrt{2}}{2}

CD=BD=\sqrt{OA^{2}+OB^{2}}=\sqrt{a^{2}+b^{2}}=\sqrt{a^{2}+\frac{a^{2}}{2}}=a\sqrt{\frac{3}{2}},

OC=\sqrt{OD^{2}-CD^{2}}=\sqrt{2a^{2}-\frac{3}{2}a^{2}}=\frac{a\sqrt{2}}{2}.

\cos\gamma=\cos\angle=\frac{OC}{OD}=\frac{\frac{a\sqrt{2}}{2}}{a\sqrt{2}}=\frac{1}{2}.

\gamma=60^{\circ}

\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma=1

\cos\gamma=\sqrt{1-\cos^{2}\alpha-\cos^{2}\beta}=\sqrt{1-\cos^{2}45^{\circ}-\cos^{2}60^{\circ}}=\sqrt{1-\frac{1}{2}-\frac{1}{4}}=\frac{1}{2}.

\gamma=60^{\circ}