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Although normally it is the line that is considered to be tangent to an arc, an arc can be tangent to infinitely many lines and so the answer to the question is: in infinitely many ways.
I can't answer it
Nothing "happens". They meet, and the rays continue merrily along their ways, never to meet again!
The tangent ratio is defined in several different ways. One of these consists of infinite series: the series for the tangent function contains some coefficients which are difficult to calculate. However, the tangent series van be easily calculated from the sine and cosine ratios.sin(x) = x - x3/3! + x5/5! - x7/7! + ...andcos(x) = 1 - x2/2! + x4/4! - x6/6! + ...where x is the angle defined in radians, and n! represents 1*2*3*...*n.So tan(x) = sin(x)/cos(x).None of these require a right angle.
They are all ways of representing numbers.