Answer
$$
\begin{aligned}
& M_R=64.0 \mathrm{lb} \cdot \mathrm{ft} \\
& \alpha=94.7^{\circ} \\
& \beta=13.2^{\circ} \\
& \gamma=102^{\circ}
\end{aligned}
$$
Work Step by Step
$$
\begin{aligned}
\mathbf{M}_1 & =40 \cos 20^{\circ} \sin 15^{\circ} \mathbf{i}+40 \cos 20^{\circ} \cos 15^{\circ} \mathbf{j}-40 \sin 20^{\circ} \mathbf{k} \\
& =9.728 \mathbf{i}+36.307 \mathbf{j}-13.681 \mathbf{k} \\
\mathbf{M}_2 & =-30 \sin 30^{\circ} \mathbf{i}+30 \cos 30^{\circ} \mathbf{j} \\
& =-15 \mathbf{i}+25.981 \mathbf{j} \\
\mathbf{M}_R & =\mathbf{M}_1+\mathbf{M}_2=-5.272 \mathbf{i}+62.288 \mathbf{j}-13.681 \mathbf{k} \\
M_R & =\sqrt{(-5.272)^2+(62.288)^2+(-13.681)^2}\\&=63.990\\&=64.0 \mathrm{lb} \cdot \mathrm{ft} \\
\alpha & =\cos ^{-1}\left(\frac{-5.272}{63.990}\right)=94.7^{\circ} \\
\beta & =\cos ^{-1}\left(\frac{62.288}{63.990}\right)=13.2^{\circ} \\
\gamma & =\cos ^{-1}\left(\frac{-13.681}{63.990}\right)=102^{\circ}
\end{aligned}
$$
