In Pursuit of the Unknown Read Free Page A

direct applications may seem limited. However, any triangle can be cut into two right-angled ones as in Figure 6 (page 11), and any polygonal shape can be cut into triangles. So right-angled triangles are the key: they prove that there is a useful relation between the shape of a triangle and the lengths of its sides. The subject that developed from this insight is trigonometry: ‘triangle measurement’.
    The right-angled triangle is fundamental to trigonometry, and in particular it determines the basic trigonometric functions: sine, cosine, and tangent. The names are Arabic in origin, and the history of these functions and their many predecessors shows the complicated route by which today’s version of the topic arose. I’ll cut to the chase and explain the eventual outcome. A right-angled triangle has, of course, a right angle, but its other two angles are arbitrary, apart from adding to 90°. Associated with any angle are three functions, that is, rules for calculating an associated number. For the angle marked A in Figure 5 , using the traditional a , b , c for the three sides, we define the sine (sin), cosine (cos), and tangent (tan) like this:
    sin A = a / c cos A = b / c tan A = a / b
    These quantities depend only on the angle A , because all right-angled triangles with a given angle A are identical except for scale.

    Fig 5 Trigonometry is based on a right-angle triangle.
    In consequence, it is possible to draw up a table of the values of sin, cos, and tan, for a range of angles, and then use them to calculate features of right-angled triangles. A typical application, which goes back to ancient times, is to calculate the height of a tall column using only measurements made on the ground. Suppose that, from a distance of 100 metres, the angle to the top of the column is 22°. Take A = 22° in Figure 5 , so that a is the height of the column. Then the definition of the tangent function tells us that
    tan 22° = a /100
    so that
    a = 100 tan 22°.
    Since tan 22° is 0.404, to three decimal places, we deduce that a = 40.4 metres.

    Fig 6 Splitting a triangle into two with right angles.
    Once in possession of trigonometric functions, it is straightforward to extend Pythagoras’s equation to triangles that do not have a right angle. Figure 6 shows a triangle with an angle C and sides a, b, c . Split the triangle into two right-angled ones as shown. Then two applications of Pythagoras and some algebra 4 prove that
    a 2 + b 2 − 2 ab cos C = c 2
    which is similar to Pythagoras’s equation, except for the extra term −2 ab cos C . This ‘cosine rule’ does the same job as Pythagoras, relating c to a and b , but now we have to include information about the angle C .
    The cosine rule is one of the mainstays of trigonometry. If we know two sides of a triangle and the angle between them, we can use it to calculate the third side. Other equations then tell us the remaining angles. All of these equations can ultimately be traced back to right-angled triangles.
    Armed with trigonometric equations and suitable measuring apparatus, we can carry out surveys and make accurate maps. This is not a new idea. It appears in the Rhind Papyrus, a collection of ancient Egyptian mathematical techniques dating from 1650 BC. The Greek philosopher Thales used the geometry of triangles to estimate the heights of the Giza pyramids in about 600 BC. Hero of Alexandria described the same technique in 50 AD. Around 240 BC Greek mathematician, Eratosthenes, calculated the size of the Earth by observing the angle of the Sun at noon in two different places: Alexandria and Syene (now Aswan) in Egypt. A succession of Arabian scholars preserved and developed these methods, applying them in particular to astronomical measurements such as the size of the Earth.
    Surveying began to take off in 1533 when the Dutch mapmaker Gemma Frisius explained how to use trigonometry to produce accurate maps, in Libellus de