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Chapter 6: Problem 24

If \(\sec x=-4,\) find \(\sec (-x)\)

Short Answer

Expert verified
\(\sec(-x) = -4\) because secant is an even function.

Step by step solution

01

Understanding Secant Function Property

Recall that the secant function is even, meaning that \( \sec(-x) = \sec(x) \). This property of even functions is crucial to solving the exercise.
02

Apply Property to Given Function

Given that \(\sec x = -4\), by the even property, we have \(\sec(-x) = \sec x\). Substitute the known value to find \(\sec(-x) = -4\).

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Key Concepts

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

Secant Function
The secant function, denoted as \( \sec x \), is closely related to the cosine function. It is defined as the reciprocal of the cosine function: \( \sec x = \frac{1}{\cos x} \). This means wherever the cosine of an angle is zero, the secant is undefined because division by zero is not possible.
Secant allows us to explore relationships within a triangle, particularly in right triangles. It helps find side lengths when different angles are given, analogous to its reciprocal, cosine. When working with secant, remember:
  • Secant is undefined where cosine is 0, such as at angles \( \frac{\pi}{2} \), \( \frac{3\pi}{2} \), etc.
  • In terms of graph symmetry, secant shares some properties with cosine, especially regarding period and amplitudes.
  • Like cosine, secant has a repeated value every \(2\pi\) due to periodicity inherent in trigonometric functions.
Understanding these aspects allows you to know where secant applies in solving problems related to angles and triangle side lengths.
Even Function
An even function is symmetrical about the y-axis. This means for an even function \(f\), it holds that \(f(x) = f(-x)\) for all values of \(x\). One clear example of an even function is the cosine function, and by extension, the secant function since \(\sec x = \frac{1}{\cos x} \). This symmetry makes calculations easier because it provides a direct relationship between positive and negative values of the same input.
Even functions like secant simplify solving problems related to trigonometry:
  • They allow us to predict the behavior of the function on both sides of the y-axis.
  • With \(\sec(-x) = \sec(x)\), you immediately know the value of the secant function for negative inputs by looking at the positive input counterpart.
This inherent symmetry is what allows us to make quick calculations, especially when dealing with unknown angles.
Properties of Functions
Understanding the properties of functions in trigonometry is essential for solving problems efficiently. For trigonometric functions like secant, these properties can include periodicity, symmetry (even or odd nature), and their undefined points.
Here's why these properties matter:
  • Periodicity: This means the function repeats its values in regular intervals, which helps predict future values.
    For secant, this period is \(2\pi\), matching that of cosine.
  • Symmetry: Knowing a function's symmetry can immediately provide answers, as with even functions like secant.
  • Undefined Points: These occur where the denominator of its reciprocal function (cosine for secant) is zero.
    For secant specifically, this means it is undefined at \(\frac{\pi}{2} + k\pi\), where \(k\) is an integer.
These properties ensure clarity and speed when dealing with trigonometric problems, making function behavior more predictable and manageable.

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