Answer: 777

Small Hint:

Compute $BD^2$ by the Law of Cosines in triangles $ABD$ and $CDB,$ and set the results equal

Big Hint:

Because $AD \ne BC,$ dividing by $AD - BC$ leaves $\cos A = \frac{AD + BC}{360},$ and $AD + BC = 640 - 360$

Solution:

Let $\angle A = \angle C = \alpha,$ $AD = x,$ and $BC = y.$ Applying the Law of Cosines to diagonal $BD$ in triangles $ABD$ and $CDB,$ $$\begin{aligned} BD^2 &= x^2 + 180^2 \\ &\quad {}- 2 \cdot 180x\cos\alpha \\ &= y^2 + 180^2 \\ &\quad {}- 2 \cdot 180y\cos\alpha. \end{aligned}$$

Rearranging gives $x^2 - y^2 = 2 \cdot 180(x - y)\cos\alpha,$ and since $x \ne y$ we may divide by $x - y:$ $$\begin{aligned} \cos\alpha &= \frac{x + y}{360} \\ &= \frac{640 - 2 \cdot 180}{360} \\ &= \frac{280}{360} = \frac{7}{9}. \end{aligned}$$

Then $1000\cos A = \frac{7000}{9} = 777.7\ldots,$ so $\lfloor 1000\cos A \rfloor = 777.$