Answer
The third smallest $\alpha$ is $~~\alpha = 7.725$
Work Step by Step
In part (a) we found that $~~tan~\alpha = \alpha$
We can use trial and error to find the third smallest value of $\alpha$:
When $\alpha = 5:~~$ then $~~tan~5 = -3.38$
When $\alpha = 6:~~$ then $~~tan~6 = -0.291$
When $\alpha = 7:~~$ then $~~tan~7 = 0.871$
When $\alpha = 8:~~$ then $~~tan~8 = -6.80$
When $\alpha = 7.5:~~$ then $~~tan~7.5 = 2.706$
When $\alpha = 7.8:~~$ then $~~tan~7.8 = 18.507$
When $\alpha = 7.7:~~$ then $~~tan~7.7 = 6.44$
When $\alpha = 7.75:~~$ then $~~tan~7.75 = 9.58$
When $\alpha = 7.72:~~$ then $~~tan~7.72 = 7.42$
When $\alpha = 7.73:~~$ then $~~tan~7.73 = 8.02$
When $\alpha = 7.725:~~$ then $~~tan~7.725 = 7.710$
When $\alpha = 7.726:~~$ then $~~tan~7.726 = 7.771$
When $\alpha = 7.7255:~~$ then $~~tan~7.7255 = 7.74$
To three decimal places, the best estimate is $~~\alpha = 7.725$
The third smallest $\alpha$ is $~~\alpha = 7.725$
