How to find tangential and normal acceleration?

[reslion]

Homework Statement

The problem asks for the tangential and normal acceleration of the acceleration. We were given that:
$$a_x=c*cos[d*t]$$ and $$a_y=c*sin[d*t]$$ where c and d are constants.

Homework Equations

The book gives us
$$a_t=(r\ddot{\theta}+2\dot{r}\dot{\theta})$$, (1)
$$a_n=(\ddot{r}-r\dot{\theta}^2$$ (2)
and
$$a=\sqrt{(a_t)^2+(a_n)^2}$$ (3)
but I found online that
$$a_t=\frac{dv}{dt}|v|$$ (4).
Finally, I know that $$a=\sqrt{(a_x)^2+(a_y)^2}$$ (5).

The Attempt at a Solution

My attempt was to use equation (4) as we don't have a theta to find $$a_{tx}$$ and $$a_{ty}$$. But wouldn't that just be the given accelerations as we would integrate the acceleration just to take the derivative again?
After that I would just solve (3) for $$a_n$$ of x and y and then plug both it in the (5) to find the magnitude of both $$a_t$$ and $$a_n$$. But I'm confused on how to find the tangential with either (1) or (4)?

 

[haruspex]

I don't understand your equation 3. A correct version of that would be useful.
Remember that the tangential acceleration is, by definition, the component of the acceleration in the direction of the velocity.