1 {photo of a bridge construction showing a latticework of triangles}
1 Caption: Triangles

Triangle. Public domain image from Nordisk familjebok.

What is the loudest instrument in a classical orchestra? Probably not a woodwind. Trumpet maybe? Or the timpani? Someone once told me - I forget if it was a music teacher or some TV show which possibly some of you may have also seen - that the loudest instrument is... the triangle.

Maybe not "loud" in the usual sense, but the triangle is certainly one of the most piercing and easily audible instruments when played in concert with the rest of the orchestra. Listen specifically for a clarinet or a cello or even a French horn and they occasionally get drowned out by the mass of instruments playing around them. But listen for the triangle and you can pick out every single time it is struck, no matter how loud all of the other instruments are playing. That ringing sound is so distinctive that it was used as the closing note in the "meet the orchestra" segment of the Disney classic Fantasia.

Triangles are very important shapes, and not only for their musical qualities. Take three lengths of rigid material. As long as one of them is not longer or equally as long as the other two put end to end, you can form them into a triangle. You can form them into a triangle - just one. There is no choice. You can't decide to make one of the angles a bit bigger or a bit smaller, because then the pieces won't join at their ends.^[1]

This property is very useful in construction and engineering. If you make a framework out of some building material, be it wood or steel beams or something else, you're best off making it out of connected triangles. Take a construction toy like Meccano or LEGO Technic or Tinkertoys or whatever and make a big open rectangle shape. If you push two diagonally opposite corners together, the shape will deform fairly easily, becoming a non-rectangular parallelogram, possibly slightly bending the components in the process. The rectangle shape is loose and sloppy. Not at all what you want in constructing something like a large building or a transmission tower.

The Eiffel Tower is made of a lattice of triangles (and some decorative elements).

Now brace the diagonal with another building element, cutting the rectangle into two triangles. The shape becomes much more stable. The diagonal piece resists stretching or squashing, keeping the diagonal to a fixed length, which keeps the entire rectangle in shape. This is because the triangles are made of fixed lengths of material, and when you have three fixed lengths you can only assemble them into one specific triangle; there's no allowance for the pieces to join at any other angles.

(In reality our buildings do have lots of rectangular features, like doors and windows, because we like rectangles and they make certain architectural features easier to build. But behind the scenes, hidden inside the walls, there are often triangular elements doing the heavy work of holding the building in shape. Not all buildings - some are built of rigid enough materials, such as bricks and concrete, that rectangles are stable enough. For less rigid exposed structures like transmission towers, you can see the triangle elements clearly.)

So if you have a triangle and you know the lengths of all three sides, the angles at the corners are determined. You can work out what they are, measured in degrees (or any other measure of angle, such as radians or gradians). The practice of working out all of the lengths and angles in a triangle is known as solving the triangle. And to solve a triangle you use the branch of mathematics known as trigonometry, from the Greek trigonon, meaning "triangle", and metron, meaning "measure".

Triangular sign, not triangular sine.

Trigonometry is built on the foundation of geometry, and is expressed through the use of two fundamental functions: the sine function and the cosine function. (These two functions are actually connected by a simple relationship, but it's easier to start talking about them separately.) These are functions of angle; each one takes an angle as input and returns a number.

What units are the angles measured in? It doesn't matter. They are functions of the physical angle, not the number used to measure the angle. Similarly, imagine a function which tells you how long it takes to drive a certain distance at, say, a fixed speed of 60 km/h. If the distance is 60 km, then it takes one hour. But if the distance is 37.28 miles, then it takes... one hour, because it's actually the same physical distance, just measured in different units. The number attached to the distance doesn't matter; what matters is how long the distance actually is. The same with the sine and cosine functions. The sine of 45° and the sine of π/4 radians are the same number, because 45° and π/4 radians are the same physical angle, just measured in different units of angle.

So what are the sine and cosine functions? Imagine a clock face, with a second hand sweeping out the seconds. Let's measure the angle that the second hand makes from the 12 o'clock position as it sweeps around a full minute of time. Let's also measure the position of the tip end of the second hand as it moves around dial. We'll measure that by measuring the distance the tip is to the right of the centre of the clock (call that the x direction), and the distance the tip is above the centre of the clock (the y direction). What distance units will we use? Let's say the length of the second hand is exactly one unit. If it's a metre long, we'll measure the x and y coordinates of the tip in metres; if it's 23 cm long, we'll measure in multiples of 23 cm.

Clock face showing x and y distances when a hand is at angle 45° from 12 o'clock.

The second hand starts its minute-long journey pointing straight up at the 12. The angle between the 12 and the second hand is zero. The tip's x distance is also zero. But the tip's y distance is one unit; the tip is exactly the length of the second hand above the centre of the clock.

The second hand sweeps a circle around the clock face, and after fifteen seconds it's pointing at the 3 o'clock position. The angle between the 12 and the second hand is now 90°. The tip's x distance is now one unit. And the tip's y distance is now zero. Similarly, we can look at the position after 30 seconds (angle 180°, x distance zero, y distance minus one unit, i.e. one unit below the centre of the clock), after 45 seconds (angle 270°, x distance minus one unit, y distance zero), and 60 seconds (back where we started).

What about the in-between angles? Let's start with 45°, which is halfway between the 1 and the 2 o'clock positions, and occurs when the second hand has been moving for 7.5 seconds. Here, the tip of the second hand is exactly as far right of the clock centre as it is above it; in other words the x and y coordinate distances are the same. The second hand itself forms the diagonal of a triangle, with the x and y distances making two other sides which meet at a right angle. Knowledge of Pythagoras' theorem for right-angled triangles and the fact that the second hand is one unit long lets us calculate that the x and y distances must both be equal to (one divided by the square root of 2) units, or approximately 0.7071 units. Interestingly, although the angle 45° is half the angle 90°, the x and y distances of the tip of the second hand are not halfway between their values at 12 o'clock and 3 o'clock.

In a similar way, we can look at what happens when the second hand is pointing at the 1 o'clock position. This is an angle of 30°. The triangle formed by the second hand and the x and y distances can be solved by noticing that it is exactly half of an equilateral triangle, and that the x distance must therefore be exactly half the length of the second hand, i.e. 0.5 units. Pythagoras again then gives us the y distance, which turns out to be (the square root of 3)/2 units, or about 0.8660 units.

Figure 1: Plots of x distance (red) and y distance (blue) of tip of a clock hand versus angle. Adapted from Creative Commons Attribution image by Wikimedia user Geek3.

There are a few more angles we can solve using geometry, but in fact almost all of the angles cannot be solved in this way. We could however simply measure the x and y distances with an accurate ruler to get an idea of the sort of values they take on as the second hand sweeps around the clock. If we do this for a full circle, we get two functions that look like the ones shown in Figure 1. The red line shows the x distance of the end of the second hand as it sweeps through a full 360° around the clock, while the blue line shows the y distance.

These two functions are exactly the sine function and the cosine function, respectively! The sine function is a function of angle that starts at 0 when the angle is 0°, rises up to a gentle peak at 1 when the angle is 90°, goes back down to 0 at 180°, falls to a minimum value of -1 at 270°, then returns to 0 at the full 360°, ready to start another cycle. I won't describe the cosine function - you can see for yourself how it behaves with the blue line. These two functions are all you need to describe the position of the end of a clock hand as it moves around the clock. The x position of the end is given by sin(angle), and the y position by cos(angle), where sin and cos are the standard abbreviations for the sine and cosine functions.

(In practice, for various reasons, mathematicians prefer to define angles around the circle starting at the 3 o'clock position and increasing anticlockwise. If you use this definition rather than the clock definition, then the x position of the end is given by cos(angle), and the y position by -sin(angle). So if you read any mathematics texts on this topic, you'll usually see the functions given in that order.)

Statue of Pythagoras making a right angle triangle, on Samos. Creative Commons Attribution-Non Commercial image by Anders Gustavson.

Okay, so we know about the sine and cosine functions. But what good are these functions, and what do they have to do with triangles? We got a bit of a hint of that when figuring out the exact values for the x and y distances at angles of 45° and 30°. We calculated them using right-angled triangles and Pythagoras - some fairly basic geometry. Think about the 45° right-angled triangle for a minute. If we know the long diagonal length (usually called the hypotenuse) of a right-angled triangle is one unit, and we know one of the other angles is exactly 45°, then we know the lengths of the other two sides are 1/sqrt(2) = 0.7071... units long, or in other words sin(45°) units long. And now think about the 30° triangle formed when the clock hand was pointing at 1 o'clock. If we know the hypotenuse of a right-angled triangle is one unit in length, and we know one of the other angles is exactly 30°, then we know that the shortest side of the triangle (the x distance in our clock) is 1/2 a unit long, and the other side (the y distance) is sqrt(3)/2 = 0.8660... units in length.

Extending this... think about a right-angled triangle where we have measured one of the other angles. It could be any angle at all, between the values of 0° and 90° (not inclusive). Call this angle θ (the Greek letter theta). Place this known angle θ at the centre of the clock face, with the hypotenuse being the clock hand. If we know the long diagonal is exactly one unit in length, then the length of the side of the triangle opposite the known angle - i.e. the x distance - is equal to sin(θ). And the length of the remaning side of the triangle - i.e. the y distance - is equal to cos(θ).

Now we can discard the clock face. Take any triangle with a right angle in it, and measure one of the other angles. Whatever that angle turns out to be, label it θ. Now, if the hypotenuse of the triangle (i.e. the side opposite the right angle) is one unit long, then the length of the side opposite the angle θ is sin(θ), and the length of the remaining side is cos(θ)!

That seems pretty handy, but how often do you run across triangles where the length of the long side is exactly one unit of length? This is where the trick of not specifying the unit of length comes in. If you have a right-angled triangle and you measure the long side (hypotenuse) as being 87 centimetres, then all is not lost! Because you can make "one unit of length" equal to 87 centimetres. Then once you've measured you angle θ, the length of the side opposite the angle θ is sin(θ) units of length, or in other words, sin(θ) times 87 centimetres! And similarly the length of the remaining side of the triangle is cos(θ) times 87 centimetres.

And this is how the sine and cosine functions are used in their most basic application. If you have a right-angled triangle, and you measure any one of the side lengths, and any one of the other angles, then using the sine and cosine functions you can solve the triangle - work out the lengths of all the sides. (If you measure a side other than the hypotenuse, you can form a simple equation like (known length) = sin(θ)*(unknown hypotenuse) and solve for the hypotenuse.) And right-angled triangles are actually pretty common. Our obsession with rectangular structures means that there are right angles all over the place, from the angles joining walls to one another and to floors and ceilings, to the angles at the corners of many property boundaries.

Trig point on summit of Mt Canobolas. This marker can be sighted by surveying equipment from tens of kilometres away, and is used for surveying large distances over much of central New South Wales.

One of the major uses of basic trigonometry is in surveying - figuring out the distances between objects and boundaries across the land. Rather than measure the distances between distant objects directly, which would involve a lot of travel, you can measure one distance between two visible objects (called the baseline), and then measure the angles between that line and other objects. This allows you (with slightly more complicated calculations than explained here) to work out the distances to any object you can see, without actually having to travel there and measure it directly. Special highly visible markers called triangulations stations are even erected across the landscape to enable this.

You can even apply trigonometry to much greater distances, such as measuring the distance from Earth to nearby stars. The size of the Earth's orbit is known, and it travels around it once every year. This means that if you take measurements of the angle formed by looking at stars six months apart, you can have a baseline of twice the size of the Earth's orbit. The angles to nearby stars can be measured by comparing their positions in the sky to objects much further away (in practice, most stars are far enough away to be considered "much" further away). Once you have the angles, you can use trigonometry to calculate the distance to the star. The angles involved are very small, and we can only measure them accurately enough to do this for a handful of the nearest stars, but it is enough to give us very confident measures of the enormous distances to these stars.^[2]

There are many, many more uses for trigonometry. I have barely even scratched the surface here. I use trigonometry almost every day at my work, which involves optical physics and image processing. Virtually every technical field uses it heavily, from archaeology and palaeontology to computer graphics and video game programming. Once you've learnt arithmetic and enough algebra to solve a simple equation, trigonometry is probably the next most practical and useful part of mathematics that you can learn. Which is why it's such a big part of high school mathematics programs (or if it isn't in some places, why it should be), even though students have difficulty seeing the usefulness of it at the time. I know I did, but I also know that now I'm glad it was drummed into me. And whenever I hear the clear, ringing note of the triangle in an orchestra, it's a reminder of how important and cool triangles are.

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[1] I'm talking about somewhat idealised lengths of material here. If you take actual lengths of wood or something, then you can fiddle things a little, depending on how loose your definition of "joined at their ends" is, if you're allowed to shave bits off to make the joins neat, if you connect them with nails or glue, or if you allow the wood to bend a bit.

[2] Our current best instrument for this was the European Space Agency's Hipparcos satellite, which measured the positions of roughly 100,000 stars during its operational lifetime. This allowed trigonometric parallax measurements to the distances of about 10,000 nearby stars with uncertainties better than 10%. ESA's replacement satellite Gaia, scheduled to be launched late in 2013, is expected to expand these measurements to approximately one billion (a thousand million) stars!