Knowing the base angles of a trapezoid is not sufficient to find its area. You would need its height or length of one slope and the length of its base.
Area=½h(b₁+b₂) where h=height and b₁ and b₂ are the lengths of the parallel sides (bases). If a slope, length s₁, makes an angle θ₁ with a base, length b₁ are known, then h=s₁sinθ₁. But we need to know either the other slope length s₂ or the angle θ₂ to find b₂, because we already have h.
h=b₂sinθ₂, so b₂=h/sinθ₂ if θ₂ is known, or sinθ₂=h/b₂. That is, b₂=s₁sinθ₁/sinθ₂ or sinθ₂=s₁sinθ₁/b₂.
Assuming b₁ is the longer base, b₂=b₁-(s₁cosθ₁+s₂cosθ₂). Or, if b₂ is known, b₁=b₂+s₁cosθ₁+s₂cosθ₂.
The formula becomes: area=½s₁sinθ₁(b₁+b₂), which becomes:
area=½s₁sinθ₁(2b₂+s₁cosθ₁+s₂cosθ₂) or
area=½s₁sinθ₁(2b₁-(s₁cosθ₁+s₂cosθ₂)).