Calculating filter parameters from system response
Just math I think. Let's substitute g = ALPHA -1, which makes the algebra a little easier. Then you need to solve 
$$\left | \frac{1+g}{1+g\cdot e^{-j\omega }} \right |^{2}=0.5$$
That means
$$\frac{1+g}{1+g\cdot e^{-j\omega }}\cdot \frac{1+g}{1+g\cdot e^{+j\omega }} = 0.5$$
Multiplying this out yields
$$\frac{(1+g)^{2}}{1+2\cdot g\cdot cos(\omega) + g^{2} }= 0.5$$
We can put this into standard quadratic equation form:

$$g^{2}+(4-2\cdot cos(\omega))\cdot g + 1 = 0$$

Solve, plug in the numbers and you get
$$g_{1} = -1.0744, g_{2} = -0.93074$$

The first solution doesn't make sense since it would be a pole outside the unit circle, so the second is correct and we get the expected

$$\alpha = 0.069262$$