The Cotangent Function

Cotangent is defined as cot(theta) = cos(theta)/sin(theta). Since sine sits in the denominator, cotangent is undefined exactly where sine equals zero: at theta = 0, π, 2pi, and so on. At each of those points the graph has a vertical asymptote. Between two consecutive asymptotes, cotangent does not repeat tangent's climb, it slides downhill. Near the left asymptote, sine is a tiny positive number and cosine is close to 1 or -1, so cotangent shoots up toward positive infinity; near the right asymptote sine shrinks toward zero from the other side and cotangent plunges toward negative infinity. Tangent does the opposite because its denominator is cosine, not sine, so its asymptotes and its uphill climb happen at different places.

Graph y = cot(x) from 0 to 2pi and you get downward-sloping branches on (0, π) and (π, 2pi), each crossing the x-axis at π/2 and 3pi/2, where cosine is zero. Lay y = tan(x) beside it and the two graphs look like mirror images shifted by a quarter period, climbing where cotangent falls, with asymptotes swapped between multiples of π and odd multiples of π/2.

This pattern is the key to all three reciprocal and quotient functions at once. Secant, like tangent, has cosine in its denominator, so both break wherever cosine is zero. Cosecant, like cotangent, has sine in its denominator, so both break wherever sine is zero. Every asymptote on every one of these graphs traces straight back to a zero of sine or cosine, which is why this whole family of functions reads as one idea wearing four outfits.