Problem 101
Question
Use the fact that \(\sec x=\frac{1}{\cos x}\) to explain why the maximum domain of \(y=\sec x\) consists of all real numbers except odd integer multiples of \(\pi / 2\).
Step-by-Step Solution
Verified Answer
The domain of \(y=\sec x\) is all real numbers except odd multiples of \(\pi/2\) due to cosine being zero there.
1Step 1: Understand the Relationship Between Secant and Cosine
The secant function, denoted as \(\sec x\), is defined as the reciprocal of the cosine function, which can be expressed as \(\sec x = \frac{1}{\cos x}\). This means that wherever cosine is zero, secant will be undefined because division by zero is not possible.
2Step 2: Identify Points Where Cosine is Zero
The cosine function, \(\cos x\), becomes zero at odd integer multiples of \(\frac{\pi}{2}\). Mathematically, these points can be expressed as \(x = (2n + 1) \frac{\pi}{2}\), where \(n\) is any integer.
3Step 3: Define the Domain of Secant Function
Since \(\sec x = \frac{1}{\cos x}\) is undefined where \(\cos x = 0\), we exclude these points from the domain of the secant function. Therefore, the maximum domain of \(y=\sec x\) includes all real numbers except those where \(x = (2n + 1) \frac{\pi}{2}\). This ensures no division by zero occurs.
Key Concepts
Secant FunctionCosine FunctionDomain of a Function
Secant Function
The secant function is an intriguing and essential trigonometric function. It is denoted as \( \sec x \) and is defined as the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \). The secant function is related to the hypotenuse over the adjacent side of a right triangle in terms of angles. But its importance extends beyond basic geometry.
The secant function is particularly useful because it helps in solving problems involving angles and distances. A key aspect of the secant function is that it is undefined wherever the cosine function is zero. This is because dividing by zero is mathematically inadmissible. So, understanding where \( \cos x = 0 \) gives us insight into the behavior and limitations of \( \sec x \). This relationship is foundational in grasping more complex trigonometric concepts.
The secant function is particularly useful because it helps in solving problems involving angles and distances. A key aspect of the secant function is that it is undefined wherever the cosine function is zero. This is because dividing by zero is mathematically inadmissible. So, understanding where \( \cos x = 0 \) gives us insight into the behavior and limitations of \( \sec x \). This relationship is foundational in grasping more complex trigonometric concepts.
Cosine Function
The cosine function, symbolized as \( \cos x \), is one of the primary trigonometric functions. In a right triangle, it represents the ratio of the length of the adjacent side to the hypotenuse. However, its relevance goes far beyond simple triangles.
Mathematically, the cosine function is periodic with a period of \(2\pi\). It means that its values repeat after every \(2\pi\) interval. Interestingly, the cosine function's value becomes zero at specific points. These points are crucial when we work with the secant function, as they help define where \( \sec x \) becomes undefined. Specifically, \( \cos x = 0 \) at odd integer multiples of \( \frac{\pi}{2} \), such as \( \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2} \), and so on.
Understanding these zero points allows us to determine where the secant function can't be evaluated.
Mathematically, the cosine function is periodic with a period of \(2\pi\). It means that its values repeat after every \(2\pi\) interval. Interestingly, the cosine function's value becomes zero at specific points. These points are crucial when we work with the secant function, as they help define where \( \sec x \) becomes undefined. Specifically, \( \cos x = 0 \) at odd integer multiples of \( \frac{\pi}{2} \), such as \( \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2} \), and so on.
Understanding these zero points allows us to determine where the secant function can't be evaluated.
Domain of a Function
The domain of a function is a critical concept in mathematics. It defines the set of all possible input values (or \( x \)-values) for which the function is defined. For the secant function \( y = \sec x \), the domain is a bit intricate due to its dependency on the cosine function.
Since \( \sec x = \frac{1}{\cos x} \), the secant function is undefined where \( \cos x = 0 \). Thus, to find the domain, we exclude these points. Cosine is zero at odd integer multiples of \( \frac{\pi}{2} \), i.e., \( x = (2n + 1) \frac{\pi}{2} \), where \( n \) is any integer. So, the domain of \( \sec x \) includes all real numbers except these specific values.
This exclusion ensures the function doesn't encounter division by zero, keeping the function valid and manageable.
Since \( \sec x = \frac{1}{\cos x} \), the secant function is undefined where \( \cos x = 0 \). Thus, to find the domain, we exclude these points. Cosine is zero at odd integer multiples of \( \frac{\pi}{2} \), i.e., \( x = (2n + 1) \frac{\pi}{2} \), where \( n \) is any integer. So, the domain of \( \sec x \) includes all real numbers except these specific values.
This exclusion ensures the function doesn't encounter division by zero, keeping the function valid and manageable.
Other exercises in this chapter
Problem 99
Solve each quadratic equation in the complex number system. \(2 x^{2}-3 x+2=0\)
View solution Problem 100
Solve each quadratic equation in the complex number system. \(x^{2}+x+1=0\)
View solution Problem 101
Solve each quadratic equation in the complex number system. \(-x^{2}+x+2=0\)
View solution Problem 102
Use the fact that \(\cot x=\frac{1}{\tan x}\) to explain why the maximum domain of \(y=\csc x\) consists of all real numbers except integer multiples of \(\pi\)
View solution