Non-uniform circular motion formula

I'm trying to derive the formula for the tangential acceleration of a particle undergoing circular motion [itex]a_{tan}=r\alpha[/itex] using vectors in the same way you would for uniform circular motion. [itex]r[/itex] is the radius and [itex]\alpha[/itex] is the angular acceleration.

Would it be correct to start with [itex]r(t)=cos(\omega t+\frac{1}{2}\alpha)i+sin(\omega t+\frac{1}{2}\alpha)j[/itex]? ([itex]\omega[/itex] is the angular velocity).

I just calculated the second derivative and it looks very messy at the moment and I'm trying to figure out how to reduce it to the desired equation but I'm not sure if my initial equation was correct.

Thanks