How can I find the explicit formula for the inverse of sinh?

Write y = sinh^(-1)(x) ie x = sinhy. Then writing this in terms of exponentials and multiplying by 2 we will get 2x = exp(y) - exp(-y). Multiply by exp(y) and rearrange to obtain exp(2y) - 2xexp(y) -1 = 0. Then this is is simply a quadratic in exp(y), so using the quadratic equation or completing the square we get exp(y) = x + sqrt(x^2 +1). Notice we take the positive square root since we must have exp(y) > 0. Then simply take logs of both sides to get the equation for y in terms of x. This is now the inverse of sinh.

Related Further Mathematics A Level answers

All answers ▸

Given that α= 1+3i is a root of the equation z^3 - pz^2 + 18z - q = 0 where p and q are real, find the other roots, then p and q.


Find the determinant of a 3x3 matrix.


Find the values of x where x+3>2/(x-4), what about x+3>2/mod(x-4)?


How do I integrate (sin x)^6?


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences