What's the use of trigonometry?

Trigonometry studies the properties of triangles. Once you know the laws of trigonometry, which involve the notions of sine and cosine, you can calculate practically everything you could possibly want to know about triangles. For example, if you know the lengths of two of the sides of a triangle and the size of one angle, then the laws of trigonometry enable you to work out the other sides and angles. The same is true if you know two angles and a side, or three sides.

Triangles are very simple geometrical objects and have many uses. Whenever triangles turn up, you need to know trigonometry to deal with them. Here are some examples.

Satellite navigation systems

Satellite navigation systems need to know the precise distances between points. One way of measuring the distance between two points is simply to walk, drive or fly from one to the other and measure it. But this can be time-consuming and expensive, especially if the points you're interested in lie far out in the ocean or even in space. Using a triangle is one way of getting around this problem: call the point where you (or a satellite) are observing from A and put another observer at a point B, which is a known distance d away from you. Suppose you want to know how far a third point, called C, is away from you. The points A, B and C form a triangle. You can measure the angle a between the straight line from A to C and the straight line from A to B, and tell the other observer to measure the corresponding angle b  at her end. Now you know two angles of the triangle and one side, the A to B of length d, and so you can calculate the other sides and angles, in particular the distance between you and C.

These kinds of techniques are used not only in satellite navigation, but in all kinds of navigation involving cars, ships, planes and space craft. They are used in geography, land surveying and cartography (map making). They are especially important in astronomy to measure the distance between Earth and the most distant stars.

Music

You may know from school that the graphs of the functions sin(x) and cos(x) look like waves. Sound travels in waves, although these are not necessarily as regular as those of the sine and cosine functions. However, a few hundred years ago, mathematicians realised that any wave at all is made up of sine and cosine waves. This fact lies at the heart of computer music. Since a computer cannot listen to music as we do, the only way to get music into a computer is to represent it mathematically by its constituent sound waves. This is why sound engineers, those who research and develop the newest advances in computer music technology, and sometimes even composers have to understand the basic laws of trigonometry.

But all of this doesn't only apply to sound. Waves move across the oceans, earthquakes produce shock waves and light can be thought of as travelling in waves. This is why trigonometry is also used in oceanography, seismology, optics and many other fields like meteorology and the physical sciences.

Architecture

Many modern buildings have beautifully curved surfaces. Making these curves out of steel, stone, concrete or glass is extremely difficult, if not impossible. One way around this problem is to piece the surface together out of many flat panels, each sitting at an angle to the one next to it, so that all together they create what looks like a curved surface. The more regular these shapes, the easier the building process. Regular flat shapes like squares, pentagons and hexagons, can be made out of triangles, and so trigonometry plays an important role in architecture.

Digital imaging

How can a computer generate complex images? In theory, the computer needs an infinite amount of information to do this: it needs to know the precise location and colour of each of the infinitely many points on the image to be produced. In practise, this is of course impossible, a computer can only store a finite amount of information. To make the image as detailed and accurate as possible, computer graphic designers resort to a technique called triangulation. As in the architecture example above, they approximate the image by a large number of triangles, so the computer only needs to store a finite amount of data. The edges of these triangles form what looks like a wire frame of the object in the image. Using this wire frame, it is also possible to make the object move realistically.

Digital imaging is also used extensively in medicine, for example in CAT and MIR scans. Again, triangulation is used to build accurate images from a finite amount of information. It is also used to build "maps" of things like tumours, which help decide how x-rays should be fired at it in order to destroy it.