Angle of friction:

It is the angle mode by the resultant of the limiting frictional force and the normal reaction with the normal reaction.

Let R be the resultant of $F_{max}$ and N making an angle ϕ with the normal reaction.

ϕ→Angle of friction

$R=\sqrt{F_{max}^2+N^2} =\sqrt{(μ_s N)^2+N^2} \\ \tan ϕ = \dfrac {f_{max}}N=\dfrac {μ_s N}N=μ_s$

Angle of Repose:

It is defined as the minimum angle of inclination of a plane with the horizontal for which a body kept on it will just slide down on it without the application of any external force.

$∑f_x=0 \\ μ_s N-W \sin α=0 ----(1) \\ ∑f_y=0 \\ N-W \cos α=0 ----(2) $

From (1) and (2)

$\tan α=μ \\ α=\tan^{-1} μ_s \\ \text { But } ϕ=\tan_1 μ_s \\ α=ϕ$

∴ Angle of repose = Angle of friction