## Teacher package: Trigonometry

The Plus teacher packages are designed to give teachers (and students) easy access to Plus content on a particular subject area. Most Plus articles go far beyond the explicit maths taught at school, while still being accessible to someone doing A level maths. They put classroom maths in context by explaining the bigger picture — they explore applications in the real world, find maths in unusual places, and delve into mathematical history and philosophy. We therefore hope that our teacher packages provide an ideal resource for students working on projects and teachers wanting to offer their students a deeper insight into the world of maths.

### Trigonometry

Here is all our material on trigonometry, grouped in the following categories:

• Hands-on trig: Trigonometry as we know and love it. These articles look at geometric problems concerning (mostly) triangles.
• Twisted graphs and infinite series: Treating sine and cosine as functions of a variable gives you intriguing graphs and clever calculus.
• Wave trig: To many people the word "sine" is synonymous with the word "wave". These articles look at the trig functions in their wavy capacity.

Don't forget that our sister site NRICH has many hands-on problems, activities and articles on trigonometry.

## Hands-on trig

Can triangles help spot a bomb? — Airport security staff have a daunting task. With impatient queues looming over them they need to search x-ray scans of cluttered suitcases for several items at once: knives, guns and bombs. How can we ease their task and make sure they don't miss a crucial item? To find out, scientists are trying to understand how we humans take in visual information. The humble triangle plays a crucial role in the experiments they perform.

Time and motion — Whatever is so wonderful about point B that makes all the people at point A want to get there? We sit at point C and muse on the problem using trigonometry.

Analemmatic sundials: How to build one and why they work — These instructions for building a sundial show you trigonometry in action.

Mathematical Mysteries: Trisecting the angle — A short piece on the famous problem that foxed the ancient Greeks.

The power of origami — You might not be able to trisect the angle using a straight-edge and compass, but you can do it using origami!

What is the Area of a Circle? — You might know the famous formula for an area of a circle, but why does this formula work? This explanation really is a piece of cake, served up with a hefty estimate of pi.

Defying gravity: The uphill roller — Use your trig skills to make things roll uphill!

Split reflections — Whether you are checking someone out in a mirror or bouncing the eight-ball off the cushion to win at pool, the law of reflection is one of the handiest tools you can pick up from physics. This article states the law and shows how it is sometimes broken.

Career interview: Systems engineer — Chuck Gill caught the space bug as a child when watching Alan Shepherd launch into space. Since then he's worked as a US Air Force navigator, a satellite operator, and in the US intelligence service. His work as a navigator involved more than just a bit of trigonometry.

Mathematical mysteries: Strange geometries — The angles in a triangle add up to 180°. Or do they? This article looks at strange geometries in which they don't.

Maths goes to the movies — Sine and cosine can be used to rotate things in the plane. This article explores how, with a little help from objects called quaternions, they can also be used to rotate things in 3D. And this makes them incredibly useful in creating computer generated movies.

On thin ice: The navigation toolkit — In conjunction with the Catlin Arctic Survey, Plus has produced a number of teacher toolkits, which explore the maths involved in a scientific expedition to the Arctic. The navigation toolkit has activities involving trigonometry which explore map projections.

### Twisted graphs and infinite series

Making the grade — How to calculate the derivative of f(x)=sin(x) and a monstrous example of a function made up of sine functions.

An infinite series of surprises — The trigonometric functions have interesting expressions in terms of infinite series, which helped Leonard Euler solve a tricky problem.

Beauty in mathematics — The most beautiful equation in mathematics involves the sine and cosine functions.

Imaging maths: Inside the Klein bottle — The strange figure-8 Klein bottle is the graph of a function made up of sine and cosine.

Getting a handle on soap — Mathematicians recently came up with a new way for soap to behave. Their new minimal surface is an infinitely twisted version of a plane with a handle sewn on, and it can be described in terms of trig functions.

### Wave trig

Career interview: computer music researcher — How to teach a computer to play music using sine waves.

Tsunami — The 2004 tsunami focused the world's attention on the terrifying consequences of an underwater earthquake. This article explores the underlying wave mathematics.

Forget Sudoku and smile for the camera — How to take pictures of really small things using X-rays.

Natural frequencies and music — This article looks at vibrations that can be harnessed by musical instrument makers.

A current problem — This article looks at eddies and currents, from the pungent problem of sewage outflow to the search for bodies of people who have fallen into rivers, explaining that fluid mechanics lies behind it all.

Saving lives: the mathematics of tomography — Not so long ago, if you had a medical complaint, doctors had to open you up to see what it was. These days they have a range of sophisticated imaging techniques at their disposal, saving you the risk and pain of an operation. This article looks at the maths that isn't only responsible for these medical techniques, but also for much of the digital revolution.

The dynamic sun — This article (published for the 1999 solar eclipse) explores the behaviour of our very own star, including the waves it emits.

Sine language — As an electronic musician Oli Freke has always been fascinated by sine waves, so much so that he's created a song based on them for the Geekpop festival, which is currently taking place on the Web. In this article he explores his song, touching on ancient Greek mythology, strange piano tunings and Johann Sebastian Bach.