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Chapter 1: Problem 14

Find the domain and range of the function. $$ h(t)=\cot t $$

Short Answer

Expert verified
The domain of the function \(h(t)=\cot t\) is all real numbers except integer multiples of \(\pi\), and the range of the function is the set of all real numbers.

Step by step solution

01

Determining the Domain

The cotangent function is undefined wherever its denominator, which is the sine function, is equal to zero. The sine function is zero at \(t=n\pi\), where \(n\) is an integer. Therefore, the domain of the cotangent function, \(h(t)=\cot t\), is all real numbers except integer multiples of \(\pi\), which can be represented as \(t \in \mathbb{R} - \{ n\pi : n \in \mathbb{Z} \}\).
02

Determining the Range

The range of a function consists of all possible output values. The cotangent function spans all real numbers. Thus, the range of the function, \(h(t)=\cot t\), is the set of all real numbers, represented as \(h(t) \in \mathbb{R}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cotangent Function
The cotangent function, often denoted as \( \cot \theta \), is one of the six fundamental trigonometric functions. It is the reciprocal of the tangent function, meaning it can be expressed as:\[\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\]Given this relationship, you can see that:
  • \( \cot \theta \) is undefined wherever \( \sin \theta = 0 \) because we cannot divide by zero.
  • The function captures the ratio of the adjacent side to the opposite side in a right-angled triangle, using the angle \( \theta \).
Understanding \( \cot \theta \) is critical in many areas of mathematics and physics. It's especially useful in solving triangles, modeling periodic phenomena, and analyzing wave forms.
Domain and Range
The domain and range are essential concepts when dealing with functions, including trigonometric ones like the cotangent function.For the cotangent function \( h(t) = \cot t \):
  • Domain: The function is undefined when \( \sin t = 0 \). This occurs at integer multiples of \( \pi \) (i.e., \( t = n\pi \), where \( n \) is an integer).
  • Thus, the domain is expressed as all real numbers except these points: \( t \in \mathbb{R} - \{ n\pi : n \in \mathbb{Z} \} \).
  • Range: Since the cotangent function can produce any real number, its range is unrestricted, covering the entire set of real numbers: \( h(t) \in \mathbb{R} \).
Domains help identify the possible values you can plug into a function, while ranges show what results you can expect when using real-world data in mathematical applications.
Real Numbers
Real numbers encompass all the numbers that we encounter in everyday life. They form the most widely recognized number system and include:
  • Rational numbers, such as fractions and integers.
  • Irrational numbers, like \( \sqrt{2} \) or \( \pi \), which cannot be expressed as fractions.
Real numbers are significant in defining both the domain and range of many mathematical functions, including trig functions.In the context of the cotangent function \( h(t) = \cot t \), understanding real numbers allows us to:
  • Comprehend why the function can accept all real numbers except those where it's undefined (i.e., \( t \in \mathbb{R} - \{ n\pi : n \in \mathbb{Z} \} \)).
  • Recognize why the range of the cotangent function is all real numbers, as any real value is a potential output from the function.
Thus, real numbers play a crucial role in understanding the foundation upon which many algebraic and trigonometric concepts rest.

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Most popular questions from this chapter

In Exercises \(25-34,\) find the limit. $$ \lim _{x \rightarrow(\pi / 2)} \ln |\cos x| $$

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