 #### What is trigonometry?

Trigonometry, coming from the Greek, literally means 'three corner measure', and relates to any mathematics dealing with angles and lengths in triangles. This includes the Pythagorean Theorem and trigonometric ratios/identities.

Born around 625 BC, Thales of Miletus was considered to be the founder of philosophy. During his life he spent some time in Egypt and whilst he was there it is thought that he devised an ingenious system of using shadows to measure the heights of the Pyramids.

Imagine that at a particular time of the day your shadow is 100 cm long and you are 150 cm tall. That means that the sun is shining at an angle such that the length of any shadow is two-thirds of the height of the object. So if 240 represents 2/3 of the height of the block, 120 represents 1/3 and so the height of the block must be 360 cm tall. Obviously the best time to perform this experiment would be the time in the day when the shadow length is equal to the height of the object.

Thus the system of trigonometry was born.

The Greeks realised that they could construct a set of very accurate triangles for varying angles and record the lengths of the sides opposite and adjacent to the angle in reference tables. In addition it was only necessary to do this for the unit length hypotenuse, as all other triangles are simply enlargements of this special case. And so by definition. That is, for any right-angle triangle with hypotenuse length 1 unit, we call the length of the side opposite the angle θ, sinθ (sine) and the length of the side adjacent to is called cosθ (cosine).

In a similar way, if the base length is 1 unit, we call the length of the side opposite the angle θ, tanθ (tangent). This definition becomes particularly useful when dealing with problems involving measurements of heights. For example, let us imagine we wish to measure the height of a tree and at a distance of 20 m from the base of the tree we measure an angle of 35o from the ground to the top of the tree. If the base length was 1 m, the height would be tan35o, but as the base length is 20 m, the height of the tree will be 20 × tan35o = 14.0 m (3 s.f.).

For more information on the fundamental definitions of the six trigonometric ratios, see Trigonometric Identities document.