The formula for magnetic flux is generally written as (Formula 1) $$\Phi=AB\cos\theta$$ where $B$ is the magnetic field strength, $A$ is the 'area vector' and $\theta$ is the angle between the NORMAL of the area and the magnetic field.

Is it correct if this formula is re-written as (Formula 2) $$\Phi=AB\sin\alpha$$ where $\alpha$ is the angle between the area and the magnetic field?

This ($\alpha$) is the angle that is generally given in questions and we always need to calculate ($90°-\alpha$) to get the angle $\theta$ needed for Formula 1. Since $\forall x, \cos x=\sin(90°-x)$, can the formaula for magnetic flux just be re-written in terms of this angle $\alpha$?

Unless my calculations are wrong or there is some other special reason I don't know of, would Formula 2 be correct for all applications?

The formula for magnetic flux is generally written as (Formula 1) $$\Phi=AB\cos\theta$$ where $B$ is the magnetic field strength, $A$ is the 'area vector' and $\theta$ is the angle between the NORMAL of the area and the magnetic field.

Is it correct if this formula is re-written as (Formula 2) $$\Phi=AB\sin\alpha$$ where $\alpha$ is the angle between the area and the magnetic field?

This ($\alpha$) is the angle that is generally given in questions and we always need to calculate ($90°-\alpha$) to get the angle $\theta$ needed for Formula 1. Since $\forall x, \cos x=\sin(90°-x)$, can the formaula for magnetic flux just be re-written in terms of this angle $\alpha$?

Unless my calculations are wrong or there is some other special reason I don't know of, would Formula 2 be correct for all applications?

Edit: If it is correct then is there any reason why it isn't used on formula sheets or in textbooks?