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Chapter 2: Problem 65

Write an expression for all values of \(x\) for which \(\sec x\) is undefined.

Short Answer

Expert verified
\( x = \frac{\pi}{2} + n\pi \) for any integer \( n \).

Step by step solution

01

Understanding the Secant Function

The secant function, denoted as \( \sec x \), is the reciprocal of the cosine function. Therefore, \( \sec x = \frac{1}{\cos x} \). For \( \sec x \) to be undefined, the denominator of the fraction \( \frac{1}{\cos x} \) must be zero, meaning \( \cos x = 0 \).
02

Identifying Where Cosine is Zero

The cosine function \( \cos x \) is zero wherever the angle \( x \) corresponds to an odd multiple of \( \frac{\pi}{2} \); specifically, \( x = \frac{\pi}{2} + n\pi \) where \( n \) is an integer. These are the points where the unit circle's x-coordinate is zero, making the angle \( x \) at the top and bottom of the unit circle.
03

Writing the Expression for Undefined Values

Since \( \sec x \) is undefined wherever \( \cos x = 0 \), we can write the expression for the values of \( x \) as \( x = \frac{\pi}{2} + n\pi \) for any integer \( n \).

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Key Concepts

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

Secant Function
The concept of trigonometric functions is essential in mathematics. Among them is the secant function, which holds a specific role. It is represented as \( \sec x \) and is particularly important because it gives us valuable insights about the cosine function's reciprocal. The secant function can be defined as:
  • \( \sec x = \frac{1}{\cos x} \)
  • This means it takes the reciprocal of the cosine function.
When exploring trigonometric identities and solving equations, recognizing the relationship between secant and cosine is key. Understanding this connection aids in calculating angles and comprehending the behavior of trigonometric graphs. Keep in mind that whenever the cosine function's value approaches zero, the secant function will tend to become undefined.
Undefined Values
In mathematics, understanding when a function becomes undefined is crucial. For the secant function, \( \sec x \), this occurs when the function's denominator equals zero.Let's break it down:
  • The denominator of \( \sec x = \frac{1}{\cos x} \) is \( \cos x \).
  • Therefore, \( \sec x \) becomes undefined precisely when \( \cos x = 0 \).
In simpler terms, if the cosine of an angle is zero, the secant function cannot be calculated for that specific angle. This makes those angles points of interest, as they represent discontinuities in the graph of the secant function. Knowing these points helps not only in solving trigonometric equations but also in graphing and interpreting trigonometric functions.
Cosine Zeros
The zeros of the cosine function are special angles where the cosine value is precisely zero. These points occur periodically along the trigonometric circle.To identify these places:
  • Remember that the cosine function evaluates to zero at odd multiples of \( \frac{\pi}{2} \).
  • This means the angles are: \( x = \frac{\pi}{2} + n\pi \)
  • Here, \( n \) is any integer (e.g., \(-2, -1, 0, 1, 2\), etc.).
These zeros are significant because they mark where the x-coordinate of a unit circle becomes zero. As such, they directly influence where the secant function will be undefined. Knowing these points helps when graphing and solving equations that involve trigonometric functions, ensuring a solid understanding of how sine and cosine play into these calculations.

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