Solving inclined plane using diff. frame

Question

I was trying to solve frictionless inclined plane problem using a diff. frame as shown in the figure, and can't figure out that acceleration along the plane = g.sin(Θ), but I think it should be = g/sin(Θ) as according to the figure.

I have gone through numerous examples and theory regarding such cases, but all of them use the frame with x axis along the incline plane and y axis perpendicular to it.

Answer 1

I think $\cos(90^\circ-\theta)=\frac{g}{a}$ is wrong. Angle between $\vec{g}$ and $\vec{a}$ is that, but that is not the same thing. Think about what happens when $\theta=0$ in the acceleration infinite?

To solve this problem you need to find the projection of $\vec{g}$ along the incline. See picture below that might help you.