### Introduction

This STEP support module derives some of the compound trigonometry formula ($\sin(\alpha + \beta)$ etc.) and introduces a new function. Previous assignments can be found here, but you can do this one without having done the others first.

STEP questions are difficult, they are supposed to be and you should expect to get stuck. However, as you tackle more and more STEP questions you will develop a range of problem solving skills (and spend less time "being stuck").

### About this assignment

The assignment is published as a pdf file below. Each STEP Support assignment module starts with a warm-up exercise, followed by preparatory work leading to a STEP question. Finally, there is a warm-down exercise.

The **warm up** for this assignment derives the trigonometrical identities for $\sin(\alpha \pm \beta)$ and $\cos(\alpha \pm \beta)$.

The **main STEP question (2005 STEP 2 Question 2)** introduces a function which will probably be unfamiliar to you. Don't be put off by this; the question starts by asking you to evaluate the function for some specific values so that you get a feel for what it does.

The **final question** consists of 4 questions taken from examination papers published more than 150 years ago. Needless to say, calculators should not be used!

For more on logical thinking and "if and only if" see Advanced Problem Solving Module 2 . Some more information on Euler's Totient function can be found on Wikipedia and Wolfram Mathworld (as well as many other places!).

You can explore how modular arithmetic and Euler's Totient function can be used to exchange information securely with this Nrich feature.

Can you use this diagram to derive the compound angle formulae for sine and cosine?

### Hints, support and self evaluation

The **“Hints and partial solutions for Assignment 10”** file gives suggestions on how you can tackle the questions, and some common pitfalls to avoid, as well as some partial solutions and answers.

Here is a Worked Video Solution to the STEP question from this assignment.