What Is Trigonometry? Triangles, Circles and the Waves in Everything

In a right-angled triangle, fix one of the other angles and the shape of the triangle is fixed, whatever its size, so the ratios of its sides depend only on the angle. There are three worth naming. The sine of the angle is the side opposite it divided by the hypotenuse, the longest side; the cosine is the side next to it divided by the hypotenuse; and the tangent is the opposite over the adjacent, which is also the sine over the cosine. For an angle of 30 degrees the sine is exactly a half, the cosine about 0.866 and the tangent about 0.577, and those numbers are the same for a triangle drawn on a page and one whose hypotenuse is a line of sight to the top of a mountain. Know one side and one angle and the other sides follow; know two sides and the angle follows, by the inverse functions.

That is the original use. A surveyor who knows the distance along the ground to the foot of a tower and the angle to its top has the tower's height as distance times tangent, and the ancient world used the same idea for the height of the pyramids, the width of rivers, and the distance to the Moon, which Hipparchus found in the second century BC to within a few percent. Triangulation, the method of mapping a country by measuring one baseline exactly and then only angles, chained across Britain, India and France in the eighteenth and nineteenth centuries, and the height of Everest was fixed in 1852 by angles from stations more than a hundred miles away. Some of what the triangle solves:

• •The height of anything whose base can be reached and whose top can be seen
• •The distance to anything that can be sighted from two known points, which is how the parallax of the nearest stars was found
• •The position of a ship from the angles of the sun and stars, the whole of celestial navigation
• •The forces on a slope or in a strut, resolved into their components
• •The path of a projectile and the range of a gun

Draw a circle of radius one and let a point move round it. The point's height above the centre is the sine of the angle it has turned through, and its distance to the right is the cosine, which is where the triangle definitions come from and where they stop being limited to angles under 90 degrees. As the point goes round, the sine rises from zero to one, falls through zero to minus one and returns, over and over, and a graph of it against the angle is the smooth wave that every physics textbook draws. The cosine is the same wave a quarter turn ahead. Any angle, however large, has a sine and cosine, and they repeat every 360 degrees, or every two pi radians, the unit mathematicians prefer because it measures the angle by the distance travelled round the unit circle.

A weight on a spring, a pendulum, a plucked string, an alternating current, a sound wave and a light wave all move in sines, and the reason is that the sine is the shape of the simplest possible oscillation: the motion of anything pulled back towards a centre by a force proportional to how far it has strayed. The mathematics of that, a differential equation whose solution is a sine, is the same whether the thing is a mass, a charge or a pressure. And in 1807 Joseph Fourier showed that any repeating pattern whatever, however jagged, can be built by adding sines of different frequencies, so that a square wave, a violin note or a heartbeat is a sum of sines. That theorem is what an MP3, a JPEG, a spectrum analyser, an MRI scanner and a noise-cancelling headphone all rest on, and it means that the triangle's three ratios describe not just waves but anything that can be broken into them.

The Greeks tabulated chords of circles; Indian astronomers in the fifth century switched to half-chords, which are sines, and called the half-chord jya, a bowstring. Arabic translators wrote it as jiba, which later copyists misread as jaib, meaning a fold or bay, and the Latin translators rendered that as sinus, a bay, from which the word sine comes, by way of a mistake. Cosine is the sine of the complementary angle, and tangent is the length of the line touching the unit circle. The tables that astronomers, navigators and surveyors used for a thousand years were replaced by calculators in the 1970s, and the functions themselves, keys on every scientific calculator and built into every spreadsheet, are computed by a series of additions that Newton's contemporaries worked out.

Trigonometry defines sine, cosine and tangent as the ratios of the sides of a right-angled triangle, which depend only on its angles and so let one measure heights and distances that cannot be reached, from a tower to the Moon. Extended to a point moving round a circle, the same functions become periodic waves, and because the simplest oscillation of anything is a sine and every repeating pattern is a sum of sines, they describe sound, light, tides, currents and the compression of every image and song.