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The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics (Isaac Newton and James Stirling) and reaching its modern form with Leonhard Euler (1748).
Graph trig functions (sine, cosine, and tangent) with all of the transformations. In this set of videos, we see how the line of equilibrium is affected by a vertical shift, and how the starting point is affected by a horizontal shift (phase). Shifts of graphs up and down are also called translations.