Refer to the function gd(y) = tan⁻¹(sinh y), called the Gudermannian. In a map of the earth constructed by Mercator projection, points located y radial units from the equator correspond to points on the globe of latitude gd(y).

Verify that t(y) in Exercise 49 satisfies t(y) = sec y, and find a value of a such that

Data From Exercise 49

Refer to the function gd(y) = tan⁻¹(sinh y), called the Gudermannian. In a map of the earth constructed by Mercator projection, points located y radial units from the equator correspond to points on the globe of latitude gd(y).

Let t(y) = sinh⁻¹(tan y). Show that t(y) is the inverse of gd(y) for 0 ≤ y

dt cost t(y) = = S a