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Chapter 1: Problem 19

Find the domain of the function. $$ g(x)=\frac{2}{1-\cos x} $$

Short Answer

Expert verified
The domain of the function \(g(x)=\frac{2}{1-\cos x}\) is all real numbers \(x\) excluding the values \(x = 2n\pi\) where \(n\) is an integer.

Step by step solution

01

Find where the Denominator Equals Zero

To find where the denominator equals zero, we solve the equation \(\cos x = 1\). Remember, \(\cos x = 1\) when \(x\) is a multiple of \(2\pi\). Therefore, the equation becomes \(x = 2n\pi\) where \(n\) is an integer.
02

Find the Domain of the Function

The domain of the function is the set of all real numbers excluding the solutions to the equation derived in the first step. Therefore, the domain all real numbers \(x\) such that \(x \neq 2n\pi\), where \(n\) is an integer.

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Key Concepts

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

Trigonometric Functions
Trigonometric functions are mathematical functions related to the angles of triangles and the relationships between their sides.
These include several functions such as sine, cosine, tangent, and their reciprocals.
They are fundamental in various fields, including geometry, physics, engineering, and many others.
  • They relate an angle to the ratios of two side lengths of a right triangle.
  • Trigonometric functions are periodic, meaning they repeat values in regular intervals.
For example, the sine and cosine functions have a period of \(2\pi\), and they take values from -1 to 1. This characteristic makes them ideal for modeling cyclical processes. Understanding these functions can help you analyze a wide range of real-world phenomena, from sound waves to the motion of pendulums.
Cosine Function
The cosine function is one of the primary trigonometric functions used to relate the angle of a right triangle to the adjacent side and hypotenuse. In the unit circle representation, the cosine of an angle is the x-coordinate of the point.
Some key properties of the cosine function are:
  • It is an even function, so \(\cos(-x) = \cos(x)\).
  • The period of the cosine function is \(2\pi\).
  • The range is from -1 to 1, meaning it oscillates between these two values.
In the context of finding domains of functions involving cosine, remember that certain values of cosine can lead to undefined results, especially in rational functions where the cosine appears in the denominator.
Real Numbers
Real numbers encompass all the numbers that can be found on the number line. This includes all the rational numbers (such as fractions and integers) and irrational numbers (which cannot be expressed as fractions, like \(\sqrt{2}\) or \(\pi\)).
Real numbers are used extensively in calculus and can describe quantities along a continuous line, unlike integers.
  • They allow for the precise calculation of physical phenomena, measurements, and quantities.
  • The set of real numbers is denoted by \(\mathbb{R}\).
  • In mathematical functions, the domain often consists of real numbers, unless certain values must be excluded.
Understanding real numbers helps in comprehending the scopes and limitations when defining functions for real-world scenarios.
Exclusion of Values
When determining the domain of a function, one key step is identifying any values that need to be excluded. In rational functions, this often involves ensuring that the denominator is never zero, as division by zero is undefined.
For trigonometric functions like the cosine function, find the values of the variable that make the denominator zero.
  • If the denominator involves \(\cos x\), then check for when \(\cos x = 1\), and exclude those x-values.
  • Use such findings to establish conditions for exclusion, often in terms of multiples of \(\pi\) or other constants.
In our given function \(g(x) = \frac{2}{1 - \cos x}\), the exclusion involves those angles \(x\) where \(\cos x\) equals 1, specifically the multiples of \(2\pi\), leading to a domain of all real numbers except \(x = 2n\pi\) for integer \(n\). This ensures the function remains defined for intended inputs.

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Most popular questions from this chapter

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \arcsin ^{2} x+\arccos ^{2} x=1 $$

Prove that for any real number \(y\) there exists \(x\) in \((-\pi / 2, \pi / 2)\) such that \(\tan x=y\)

Boyle's Law For a quantity of gas at a constant temperature, the pressure \(P\) is inversely proportional to the volume \(V\). Find the limit of \(P\) as \(V \rightarrow 0^{+}\).

Prove that \(\arctan x+\arctan y=\arctan \frac{x+y}{1-x y}, x y \neq 1\). Use this formula to show that \(\arctan \frac{1}{2}+\arctan \frac{1}{3}=\frac{\pi}{4}\)

Verify that the Intermediate Value Theorem applies to the indicated interval and find the value of \(c\) guaranteed by the theorem. $$ f(x)=x^{2}-6 x+8, \quad[0,3], \quad f(c)=0 $$
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