Answer
$\alpha = 4.493$
Work Step by Step
In part (a) we found that $~~tan~\alpha = \alpha$
We can use trial and error to find the second smallest value of $\alpha$:
When $\alpha = 1:~~$ then $~~tan~1 = 1.557$
When $\alpha = 2:~~$ then $~~tan~2 = -2.185$
When $\alpha = 3:~~$ then $~~tan~3 = -0.1425$
When $\alpha = 4:~~$ then $~~tan~4 = 1.1578$
When $\alpha = 5:~~$ then $~~tan~5 = -3.38$
When $\alpha = 4.5:~~$ then $~~tan~4.5 = 4.637$
When $\alpha = 4.4:~~$ then $~~tan~4.4 = 3.096$
When $\alpha = 4.45:~~$ then $~~tan~4.45 = 3.72$
When $\alpha = 4.48:~~$ then $~~tan~4.48 = 4.225$
When $\alpha = 4.49:~~$ then $~~tan~4.49 = 4.42$
When $\alpha = 4.495:~~$ then $~~tan~4.495 = 4.527$
When $\alpha = 4.494:~~$ then $~~tan~4.494 = 4.506$
When $\alpha = 4.493:~~$ then $~~tan~4.493 = 4.485$
When $\alpha = 4.4935:~~$ then $~~tan~4.4935 = 4.495$
To three decimal places, the best estimate is $~~\alpha = 4.493$
The second smallest $\alpha$ is $~~\alpha = 4.493$
