Problem 68 - Precalculus Essentials

Question

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

Step-by-Step Solution

Answer

The basic cotangent curve, \(cot(\theta)\), is defined as \(cos(\theta)/sin(\theta)\). It's reciprocal function of tangent. The function is periodic with a period of \(\pi\), and demonstrates discontinuity at \(0\) and at integer multiples of \(\pi\). For \(\theta\) in \((0, \pi)\), \(cot(\theta)\) is positive, and for \(\theta\) in \((\pi, 2\pi)\), \(cot(\theta)\) is negative. The function exhibits decreasing behavior as \(\theta\) increases.

Step 1: Defining Cotangent

Cotangent of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the opposite side, i.e. \( cot(\theta) = \frac{cos(\theta)}{sin(\theta)} \). It's also the reciprocal of tangent.

Step 2: Cotangent Periodicity

The cotangent function is periodic with a period of \(\pi\). This means that cotangent repeats its values every \(\pi\) radians.

Step 3: Cotangent Discontinuities

Cotangent is undefined wherever its denominator sin(\(\theta\)) is zero. Hence, it's undefined at \(0\) and at integer multiples of \(\pi\). These points are places where vertical asymptotes occur on the graph of cotangent function.

Step 4: Cotangent at Quadrantal Angles

At quadrantal angles, \(cot(\pi/2)\) and \(cot(3\pi/2)\) are undefined because sine is zero at these places. However, \(cot(0)\) and \(cot(\pi)\) are also undefined because we can't divide by zero.

Step 5: General Behavior of Cotangent

Generally, cotangent decreases as the angle increases within its period. For \(\theta\) in \((0, \pi)\), \(cot(\theta)\) is positive, and for \(\theta\) in \((\pi, 2\pi)\), \(cot(\theta)\) is negative.