Let's break the problem into two parts:
1) In the first part, the wheel starts with an angular speed [tex]\omega_0 = 20.0 rad/s[/tex] and it rotates with an angular acceleration of [tex]\alpha_0 = 34.0 rad/s^2[/tex] for t=1.80 s. So we can find the total angle covered by the wheel in this part of the motion:
[tex]\theta(t) = \omega_0 t + \frac{1}{2} \alpha_0 t^2 = 91.1 rad[/tex]
2) Now, the circuit breaker trips, so the wheel starts to decelerate with a certain angular acceleration [tex]\alpha_1[/tex] (which is negative). During this part of the motion, the wheel covered an angle of [tex]435 rad[/tex].
3) So, we just need to add the angles the wheel covered in the two parts of the motion:
[tex]\theta _{tot}=91.1 rad+435 rad=526.1 rad[/tex]
