Since tan θ is dimensionless, we need the RHS of the equation tan θ = v to be dimensionless.
Let the speed of the rainfall be v1. Then, by the method of dimensions, we get that tan θ = v/v1 or v1/v
When the man is stationary (v = 0), he must hold the umbrella straight.
i.e θ = 0°
tan θ = 0 when v = 0
∴ The required relation is tan θ = v/v1