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      <text>5.4.2 (History of trigonometry)</text>
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      <text><![CDATA[<h3>History of trigonometry</h3>

<p>As with much of mathematical history, early trigonometry can be attributed to the ancient Egyptians. They were primarily interested in theorems about the ratio of similar triangles, however the Egyptians didn't have a concept of the measurement of an angle. However this didn't stop them from using trigonometric ratios in spectacular fashion to build the pyramids, now around \(4000\) years old. So they must have done something right! The Rhind Papyrus, attributed to Egyptian scribe Ahmes circa (1680 - 1620 BCE) contains problems regarding the ratio of height to the side lengths of a pyramid. The solution to these problems, used a version of what we would call trigonometry today.</p>

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<p>You may be curious as to where we get the unit of degrees to measure angles. One theory says that this comes from the Babylonians (circa 300BCE). They believed that a year lasted \(360\) days and they represented a year by a circle, so they defined a circle as holding \(360^\circ\). In fact Babylonian mathematics was done to the base \(60\) (rather than base \(10\)) which is what we use today. Actually this makes a lot of sense because the number \(60\) has many factors, namely \(1,2,4,5,6,10,12, 15, 30, 60\) which make a lot of computations more straightforward.</p>
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<p>Since trigonometry is primarily concerned with finding properties of circles and triangles, it is perhaps unsurprising that it is considered a branch of geometry. In fact the name trigonometry is derived from Greek words meaning triangle and measurement. However, it is slightly different in focus to the geometry that was advanced by Euclid and the ancient Greeks in that it is primarily computational. Whereas the work of Euclid in the elements focuses primarily on geometric construction without the need for calculations at all. One of the primary applications of trigonometry for the ancient Greeks was astronomy. As the planet Earth is locally a sphere, the trigonometry used to calculate celestial distances is somewhat different than the trigonometry we have met in this course, which focusses on calculations in the plane. The geometry of spheres is called <em>spherics</em> or <em>elliptic geometry</em>. It has different rules than trigonometry in the plane (for example the angles of a triangle don't need to add to \(180^\circ\)). It seems that many of the advances in trigonometry were actually to solve problems in elliptic geometry. Much of early trigonometry is attributed to Hipparchus of Nicaea (180-125 BCE) who is sometimes called the father of trigonometry.</p>
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<p>The next major advances in trigonometry was a result of the Indian mathematicians of the fourth and fifth centuries. It was during this time that the sine function was defined as in the modern sense, as well as the cosine and inverse sine. Aryabhata (476–550CE) then tabulated many values of sine function for different degrees to astonishing accuracy. In addition the mathematicians Bhaskara and Brahmagupta produces functions that would estimate the sine values of an acute angle without resorting to a table. It was around this time the sum and difference formulas for sin became knows.</p>
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<p>As with most other knowledge, trigonometry in Europe declined during the dark ages and many techniques were lost for centuries. However in the medieval Islamic empire, many of the Indian trigonometric treatises were translated, preserved and eventually improved upon by the Islamic mathematicians. By the tenth century, Islamic mathematicians were using all of the trig functions we have met ourselves, they also began to discover more trigonometric identities. For example Abu al-Wafa discovered the double angle formula for sine, as well as the sine rule. In addition the idea of <i>triangulation</i> (this is the process of determining the location of a point by forming triangles to it from known points) was developed by Muslim mathematicians around this time. Many of these formulas were later rediscovered in Europe during the renaissance, as the continent caught up with the Islamic advances after the dark ages. Eventually Leonhard Euler would publish a paper on the sine and cosine function which led to the analytic treatment of trigonometric functions in Europe.</p>]]></text>
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