4/4/2023 0 Comments Unit circle tangentUnlike the sine and cosine however, the length of the line segment in question is not limited to values of between zero and one. It is also represented by a line segment associated with the unit circle. The tangent function, like the sine and cosine functions, is the ratio of two sides of a right-angled triangle. Note that the tangent line is perpendicular to the hypotenuse of the right angled triangle. The tangent line touches the unit circle at point P Since the unit circle has a diameter of one unit, the lengths of line segments EP and EO are also equal in value to the sine and cosine respectively. Line segments EP and EO are the opposite and adjacent sides of the right-angled triangle in relation to angle θ. Line segment OP is a radius of the circle, and also the hypotenuse of right-angled triangle OPE. The screenshot below shows the tangent line for the unit circle at the point where line segment OP intersects the circumference of the circle. More correctly, the tangent line has the same slope as the curve at the point of contact. At the point where the tangent line touches the curve, we can say that the tangent line is "going in the same direction" as the curve at that point. A tangent line is a straight line that touches a curve at a single point. The tangent function is related to the tangent line. Prior to that, the tangent and cotangent had been known by their Latin names umbra recta and umbra versa, meaning straight shadow (the horizontal shadow cast by a vertical gnomon) and turned shadow (the vertical shadow cast by a gnomon attached to a wall) respectively. Indeed, the term tangent did not appear until towards the end of the sixteenth century CE, when it was used by the Danish mathematician and physicist Thomas Fincke, in his book Geometriae Rotundi. They were familiar with all six trigonometric functions in modern use, and had compiled accurate trigonometric tables, including tables of tangents and cotangents, which they called tables of shadows. However, the relationship between the height of an object, the length of the object's shadow, and the angle of elevation of the sun in the sky does not appear to have been expressed in terms of a trigonometric function until much later.īy the middle of the ninth century CE, Arab scholars had assimilated much of the astronomical and mathematical knowledge of the ancient world. Thales recognised that, as the sun moves across the sky, the relationship between the length of an object's shadow and its height will be in the same proportion for all objects. The instrument used to perform such measurements was called a shadow stick or gnomon, one form of which can still be seen as the upright component of a sundial. The idea of using the length of shadows, both to obtain measurements of the height of physical objects and to mark time, had been used for thousands of years by the Egyptian and Babylonian civilisations. He is said to have measured the height of a pyramid, for example, by measuring the length of the pyramid's shadow (measured from the centre of the pyramid) at the same time of day that the length of his own shadow was equal to his height. During the sixth century BCE, the Greek philosopher Thales is believed to have visited Egypt, where he learned of Egyptian methods for calculating heights and distances by measuring the lengths of shadows. Whereas the sine and cosine evolved from the need to make astronomical calculations, the tangent evolved more from things like the need to calculate the height of buildings. The tangent function has a somewhat different history from the sine and cosine functions. On this page we look at the last of the three main trigonometric functions ( sine, cosine and tangent).
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