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Chapter 1: Problem 171

Find the period of \(f(x)=\tan x \cdot \cot x\).

Short Answer

Expert verified
The period of \(f(x) = \tan x \cdot \cot x\) can be argued to be \(\pi\), even though simplification of the function results in a constant value without a conventional period.

Step by step solution

01

Analyzing the function

Start by looking at the given function, \(f(x) = \tan x \cdot \cot x\). Since \(\tan x\) and \(\cot x \) are identical except that tangent is the reciprocal of cotangent, it is possible to simplify \(\tan x \cdot \cot x\) to 1.
02

Finding the period of the function

Now, the simplified function becomes \(f(x) = 1\). This function is a constant function, which means that it doesn't oscillate and thus it doesn't technically have a period. However, when the task specifically asks for the period of this combination of \(\tan x\) and \(\cot x\), an argument can be made that since \(\tan x\) and \(\cot x \) are periodic with a period of \(\pi\), the period of \(f(x) = \tan x \cdot \cot x\) is \(\pi\)as well.

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Key Concepts

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

Periodicity
In trigonometry, the concept of periodicity refers to the characteristic of a function to repeat its values at regular intervals. This makes the graph of the function look like a wave pattern that repeats itself over and over.
For most trigonometric functions, this repeating interval is called the period. For example, the sine and cosine functions have a period of \(2\pi\), meaning they repeat every \(2\pi\) units. The tangent and cotangent functions, however, have a period of \(\pi\).
  • A function is periodic if there exists a positive number \(T\), called the period, such that \(f(x + T) = f(x)\) for all \(x\) in the domain.
  • For the function \(f(x) = \tan x \cdot \cot x\), it simplifies to 1, which is a constant value.
  • Even though \(f(x) = 1\) doesn't oscillate, evaluating the original functions, \(\tan x\) and \(\cot x\), each having a period of \(\pi\) supports the understanding of its period from their properties.
Understanding periodicity helps in solving various trigonometric problems, especially when analyzing combinations of functions like in the exercise above.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved angles. They play a crucial role in simplifying expressions and solving equations that include trigonometric functions.
  • Some common identities include the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\), and reciprocal identities such as \(\tan x = \frac{\sin x}{\cos x}\) and \(\cot x = \frac{1}{\tan x}\).
  • In the given exercise, the identity used is \(\tan x \cdot \cot x = 1\), simplifying the function to a constant value.
  • These identities help in converting and reducing complex expressions or even determining the behavior of functions.
By leveraging identities, we can transform a seemingly complicated function into something much easier to work with, just like reducing \(\tan x \cdot \cot x\) to 1, which made identifying its properties straightforward.
Constant Functions
A constant function is one that returns the same value regardless of the input. It is one of the simplest types of functions and has distinctive properties.
  • The mathematical form of a constant function is \(f(x) = c\), where \(c\) is a fixed number.
  • Since the output is always \(c\), the graph is a horizontal line at \(y = c\).
  • Constant functions have no period because they don't oscillate or repeat any values at different intervals; the value remains unchanged.
In the exercise, simplifying \(\tan x \cdot \cot x\) to a constant function of \(1\) demonstrates that such functions drastically reduce complexity. Understanding this concept is vital, as it shows how combining trigonometric identities can neutralize functions into constants, which can help in problem-solving by focusing on the periodicity and identity properties ultimately simplifying the tasks.

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Most popular questions from this chapter

Find the period of the function \(f^{\prime}\) which satisfy the equation \(f(x+4)+f(x-4)=f(x)\)

Let \(f(x)=[\sin 3 x]+|\cos 6 x|\), where \([,]=\), G.I.F. Then the period of \(f(x)\) is (a) \(\pi / 3\) (b) \(2 \pi / 3\) (c) \(\pi / 6\) (c) \(\pi / 12\)

If \(f(x)\) is a polynomial of degree 2 such that \(f(0)=1\) and \(f(x+2)=f(x)+4 x+2\), find the polynomial \(f(x)\).

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\(f(x)=\left\\{\begin{array}{ll}x^{2} & : x \geq 0 \\ x & : x<0\end{array}\right.\) and \(g(x)=-|x|, x \in R\)
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