p-Trigonometric Functions

For , the inverse -sine is defined on by (Lindqvist 1995)

(1)

The -sine is the inverse function on the interval , where its quarter-period is

			(2)
			(3)
			(4)

On this interval, the -cosine is defined by . Extending the functions to the real line by reflection and periodicity gives

(5)

so parameterizes the unit superellipse . When , these definitions reduce to the ordinary sine and cosine and .

The -tangent is defined by and satisfies

(6)

Consequently, the substitution gives the improper integral

			(7)
			(8)
			(9)

where is the beta function. It follows by symmetry that

(10)

(Poodiack 2026).

The term squigonometry is used for the broader study of analogues of trigonometry associated with superellipses (Wood 2011, Poodiack and Wood 2022). These -trigonometric functions should not be confused with the area-parameterized squigonometric functions. Both systems parameterize the same unit superellipse, but the defining inverse integrands have exponents and , respectively. The -trigonometric half-period tends to 2 as , whereas the constant denoted for the squigonometric functions is the area of the unit -circle and tends to 4. The two conventions coincide when (Poodiack 2026).