The involute function is defined as

Explanation:

Involute function:

It is a function used to design profile of involute gears. This function defines tooth thickness, tooth space and other involute parameters. 

Consider the involute of a circle shown in figure (a)

l = length of unwrapped

l = AB arc = r_b (β + δ) = r_b tan α

where r_b = base radius

Thus (β + δ) = tan α

Also from figure (b)

x = r_b cosβ + l sinβ = r_b [cosβ + (β + δ)sinβ] 

y = r_b [sinβ - (β + δ)cosβ]

Also θ = α = β = tanα - δ 

θ = tanα - α - δ = inv (α) - δ

where inv (α) = tanα - α

here in this question α = ϕ

∴ Inv ϕ = tan ϕ - ϕ represents the involute function

where ϕ = pressure angle