Answer
β=45 or $\pi/4$
Work Step by Step
We have α=$\pi/3$ and γ=$2\pi/3$.
We need to find the value of β. And β is acute.
We use the Property of Direction Cosines:
$cos^2$α+$cos^2$β+$cos^2$γ=1
Put the values we have in the equation, and solve:
$cos^2$($\pi/3$)+$cos^2$β+$cos^2$($2\pi/3$)=1
$(\frac{1}{2})^2$+$cos^2$β+$(\frac{-1}{2})^2$=1
$\frac{1}{4}$+$cos^2$β+$\frac{1}{4}$=1
We sum the fractions:
$cos^2$β+$\frac{2}{4}$=1
$cos^2$β+$\frac{1}{2}$=1
Take a have from each side of the equation:
$cos^2$β=1-$\frac{1}{2}$
$cos^2$β=$\frac{2}{2}$-$\frac{1}{2}$
$cos^2$β=$\frac{1}{2}$
$\sqrt (cos^2β)$=$\sqrt (\frac{1}{2})$
$cos$β=$\frac{1}{\sqrt 2}$ and -$\frac{1}{\sqrt 2}$
Now we use the inverse function ($cos^{-1}$) to get the angle of β:
Since the angle of β is acute we use $\frac{1}{2}$
β=$cos^{-1}$$\frac{1}{2}$=$\frac{\pi}{4}$
