Problem 68 - Algebra and Trigonometry

Question

Without drawing a graph, describe the behavior of the basic cotangent curve.

Step-by-Step Solution

Answer

The cotangent curve cot(x) is defined as the reciprocal of the tangent function, so it is defined for all x except where tan(x) = 0. It is periodic with a period of π, and has a vertical asymptote at x = nπ (for n being an integer), and an x-intercept at x = (n+0.5)π. The function decreases from positive infinity to negative infinity on each interval (nπ, (n+1)π).

Step 1: Identity and Definition of Cotangent Function

The cotangent function, written as cot(x), is the reciprocal of the tangent function, that is, cot(x) = 1 / tan(x). For all real numbers x except for those where tan(x) = 0 (which are at (2n+1)π/2 for n being an integer), cot(x) is defined.

Step 2: Period of the Cotangent Function

The cotangent function is periodic, repeating its values over an interval, or a 'cycle'. The period of the basic cotangent function cot(x) is π. This means that cot(x) = cot(x + π) for any value of x where cot(x) is defined.

Step 3: x-intercepts and Asymptotes of the Cotangent Function

The cotangent function, cot(x), exhibits certain regular behavior in terms of its x-intercepts and asymptotes. It is undefined at multiples of π, so x = nπ (for n being an integer) are vertical asymptotes. Between each pair of these asymptotes, there is an x-intercept at the midpoint, that is, at (n+0.5)π.

Step 4: Behavior in Different Intervals

The behavior of the cotangent function between vertical asymptotes can be established as well. On the interval (nπ, (n+1)π) for an integer n, cot(x) decreases from positive infinity to negative infinity.