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The definition, in terms of the sides of a right angled triangle, is very restrictive since it defines tangent only for angles in the interval (0, pi/2) radians = (0, 90) degrees. However, the function is actually defined for all numbers.


So, the tangent of an angle is best defined by the ratio of the sine and cosine of the angle for any angle other than an odd multiple of pi/2.

If the angle is x radians, then

sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ...

cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ...

and then

tan(x) = sin(x)/cos(x).


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Q: What is tangent of the angle?
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