## trigonometry
"Have you any old elms?" ... ## IntroductionThe Simplon Tunnel, between Italy and Switzerland, is 20km (about 12 miles) long and was bored from both ends through the Alps. When the headings met in the middle, in 1906, they were in exact horizontal alignment and only 10cm (4in) out vertically. The engineers managed to smooth out the discontinuity. Using trigonometry they had set up their machines to cut along the 10km (6 mile) sides of two huge triangles in the mountain.## Sines, cosines and tangentsTrigonometry is the art of calculating the dimensions of triangles. The basic idea is that the ratios between the sides of a right-angled triangle depend on its base angle. The ratios have been named the sine of A(sin A), the cosine of A (cos A), the tangent of A (tan A) and others. They have been tabulated for many values of the angle A. Sin A is the length of the triangle side opposite the angle A divided by the longest side; cos A is the length of the side adjacent to the angle A divided by the longest side; and tan A is the ratio of the length of the opposite and adjacent sides of the triangle.Armed with trigonometrical tables anyone can determine the dimensions of any triangle with great accuracy. Since nearly any shape can be broken up into a series of triangles this is a powerful method of solving even complex spatial problems. To use it in tunnelling engineers set up a station from which both the ends are visible or (as this may be difficult with mountains all around) a station from which other stations are visible, from which in turn the ends can be seen. They measure the angles between all the stations by optical sighting and thus relate the two ends. Trigonometry then tells them the tunnelling angles that will align the two headings. The required accuracy of a thousandth of a degree implies a certain expertise; but the mathematical principle involved is nevertheless extremely simple. ## Trigonometry in everyday lifeTrigonometrical ratios have, however, escaped from their simple geometrical interpretation and uses in surveying and measuring, and now crop up in all sorts of mathematical problems that do not seem to be at all "angular". Some of their most fruitful applications are in circuit theory, radiation physics and information-handling, in which the angles are not real but introduced merely for convenience.The sine of 0° is 0 and it increases with increasing angles up to 90°, whose sine is 1. Between 90° and 180° the sine reduces again to 0. From 180° to 270° the sine is negative, decreasing to -1. And from 270° to 360° the sine increases again from -1 to 0. Thus if a trigonometrical angle is regarded as winding up continuously, its sine swings between +1 and -1 and back at each revolution of 360°. This periodic behaviour gives mathematicians a framework for handling waves, vibrations, oscillating radiation such as light and radio waves, and alternating current (AC) electricity. In most European countries a power station generator spins at 50 revolutions a second. As a result its output voltage (which depends on the sine of the angle of rotation) swings back and forth between positive and negative at 50 cycles per second (50Hz) to generate mains-frequency AC. Any other source of oscillation, even light with a frequency of 600 million million Hz, can be similarly assigned a notional "phase angle" winding up at the appropriate rate of time. Any vibration, however complicated, can be made up of a set of sine-wave components (or cosine-wave ones which are similar), each with its own frequency. Each frequency is quite independent of the rest. (Two stones thrown into water together generate two sets of spreading ripples which intersect and go right through each other, emerging quite unaffected.) Similarly, the human ear can pick out the notes in a chord although they make a single vibrational pattern in the air or a single groove on a gramophone record.
## Angles in a radio beamMany electronic techniques process these frequency components of vibrations in ways governed by trigonometry. An AM (amplitude modulated) radio transmitter, for example, has to take a sine-wave audio frequency A (say the musical note A, 440Hz) and attach it somehow to a radio sine-wave "carrier" C, being broadcast at perhaps one million Hz (1MHz, in the medium-wave band). It does this in effect by multiplying the audio voltage at each instant by the carrier voltage at the instant and transmitting the result. Now one of the many trigonometriical formulae for simple angles asserts that sin A × sin C = ½cos(A-C )- ½cos (A+C). Since A and C are phase-angles of an audio and carrier frequencies the result of the multiplication is two cosine-waves (just like sine-waves), one at (1,000,000 – 440) Hz and the other at (1,000,000 + 440) Hz, each of half the intensity of the original carrier.The splitting of the carrier into two closely spaced "sidebands" is called amplitude modulation, or AM. A transmission generally has many such pairs of sidebands continuously changing in their spacing and intensity with the changing frequency-components of the audio signal. At the receiver the audio signal is recovered by the reverse process of the demodulation. It may seem incredible that a mathematical formula first proved for static triangles on paper can be impudently applied to the imaginary rotating angles of an electronic signal. ## Related category• GEOMETRY | ||||||||||||||||||||||||

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