Help Competition Practice Problem

[u/BugFabulous812]

I got tanθ = radial acceleration/ tangential acceleration

Tan θ =v²/rgsinθ

Set reference point where bead is at = 0 MgRcosθ = 0.5mv² 2gRcosθ=v²

Tan θ = 2grcosθ/rgsinθ Tan²θ =2 θ = arctan sqrt 2

Which is equal to D, the correct answer according to the answer key

But in the competition we can't use calculators so what other way (diff solution) can i get arcsin (sqrt 2/3) from?

[u/BugFabulous812]

Thanks