Answer
$α = 60°$
Work Step by Step
Use the property of Direction Cosines given here:
$\cos^2 α + \cos^2 β + \cos^2 γ = 1$
Given that $β = \frac{2\pi}{3}$, $γ = \frac{\pi}{4}$ and α is acute
$\cos^2 α + \cos^2 {\frac{2\pi}{3}} + \cos^2 {\frac{\pi}{4}} = 1$
$\cos^2 α + (-0.5)^2 + ({\frac{1}{\sqrt2}})^2 = 1$
$\cos^2 α + 0.25 + 0.5 = 1$
$\cos^2 α = \frac{1}{4}$
$\cos α = +/- 0.5$
$α = 60°$
