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Chapter 5: Problem 68

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

Short Answer

Expert verified
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 蟺, and has a vertical asymptote at x = n蟺 (for n being an integer), and an x-intercept at x = (n+0.5)蟺. The function decreases from positive infinity to negative infinity on each interval (n蟺, (n+1)蟺).

Step by step solution

01

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)蟺/2 for n being an integer), cot(x) is defined.
02

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 蟺. This means that cot(x) = cot(x + 蟺) for any value of x where cot(x) is defined.
03

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 蟺, so x = n蟺 (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)蟺.
04

Behavior in Different Intervals

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

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