### Introduction

This module contains STEP 3 questions on Hyperbolic Functions.

Each STEP 3 module consists of 4 STEP questions, some topic notes and useful formulae, a "hints" sheet and a "solutions" booklet.

STEP questions are challenging, so don't worry if you get stuck. These STEP 3 modules assume that you have already begun to develop your problem-solving skills and approach to STEP questions by working on the Foundation modules and the STEP 2 modules.

### About this assignment

Hyperbolic functions are related to the Trigonometric functions, and they can be expressed rather nicely in terms of exponentials. They often appear in Calculus questions (such as Questions 1, 2 and 3 of this module).

Question 3 asks for a proof which can be done by induction. See Foundation Assignment 20 for an introduction to proof by induction. For Question 4 you need a little knowledge of complex numbers, i.e. $i^2=-1$, $\sqrt{-3}=i\sqrt{3}$ and $(a+bi)^2=a^2 - b^2 +2abi$.

You can find out more about Hyperbolic Functions on the Underground Mathematics website, and in the NRICH Advanced Problem Solving module.

You might like to look at questions 7/8, 13/14 and 15/16 from the Mixed Pure STEP III questions for some more questions on Hyperbolic Functions.

### Hints, support and self evaluation

The **“Hints"** file gives suggestions and some starting points on how you can tackle the questions. The **"Solutions"** file leads you through one possible solution, but some steps will be missing and you will still need to work through the question yourself.