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Chapter 4: Problem 68

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

Short Answer

Expert verified
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 by step solution

01

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.
02

Cotangent Periodicity

The cotangent function is periodic with a period of \(\pi\). This means that cotangent repeats its values every \(\pi\) radians.
03

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.
04

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.
05

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.

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Most popular questions from this chapter

The following figure shows the depth of water at the end of a boat dock. The depth is 6 feet at low tide and 12 feet at high tide. On a certain day, low tide occurs at 6 A.M. and high tide at noon. If \(y\) represents the depth of the water \(x\) hours after midnight, use a cosine function of the form \(y=A \cos B x+D\) to model the water's depth.

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