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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).
The tangent function. and its properties such as graph, period, phase shift and asymptotes are explored interactively by changing the parameters a, b, c and d using an applet. See figure below for main panel of the applet showing the graph of tangent function in blue and the vertical asymptotes in red. You may also want to consider another tutorial on the trigonometric unit circle. Once you finish the present tutorial, you may want to go through a self test on trigonometric graphs.