Answer
$$
\begin{aligned}
& M_R=576 \mathrm{lb} \cdot \mathrm{in} . \\
& \alpha=37.0^{\circ} \\
& \beta=111^{\circ} \\
& \gamma=61.2^{\circ}
\end{aligned}
$$
Work Step by Step
$$
\begin{aligned}
\mathbf{M}_R & =\left|\begin{array}{ccc}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
4 \cos 30^{\circ} & 5 & -4 \sin 30^{\circ} \\
0 & 0 & 60
\end{array}\right|+\left|\begin{array}{ccc}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
4 \cos 30^{\circ} & 0 & -4 \sin 30^{\circ} \\
0 & 80 & 0
\end{array}\right| \\
& =300 \mathbf{i}-207.85 \mathbf{j}+160 \mathbf{i}+277.13 \mathbf{k} \\
& =\{460 \mathbf{i}-207.85 \mathbf{j}+277.13 \mathbf{k}\} \mathrm{lb} \cdot \mathbf{i n} . \\
M_R & =\sqrt{(460)^2+(-207.85)^2+(277.13)^2}=575.85=576 \mathrm{lb} \cdot \mathrm{in} . \\
\alpha & =\cos ^{-1}\left(\frac{460}{575.85}\right)=37.0^{\circ} \\
\beta & =\cos ^{-1}\left(\frac{-207.85}{575.85}\right)=111^{\circ} \\
\gamma & =\cos ^{-1}\left(\frac{277.13}{575.85}\right)=61.2^{\circ}
\end{aligned}
$$
