What is the range of y = sec(x)?
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
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Question
no calculator way please
![The text in the image reads:
**What is the range of \( y = \sec(x) \)?**
Here is a detailed explanation for an educational website:
---
To understand the range of the secant function, \( y = \sec(x) \), let's first recall that the secant function is the reciprocal of the cosine function, \( \sec(x) = \frac{1}{\cos(x)} \).
### Key Points to Consider:
- The cosine function, \( \cos(x) \), oscillates between -1 and 1 for all \( x \).
- The secant function is undefined where \( \cos(x) = 0 \), which happens at \( x = \frac{\pi}{2} + n\pi \) for any integer \( n \).
### Analyzing the \( \sec(x) \) Function:
- When \( \cos(x) \) is positive and \( 0 < |\cos(x)| \leq 1 \), then \( \sec(x) \geq 1 \).
- When \( \cos(x) \) is negative and \( -1 \leq \cos(x) < 0 \), then \( \sec(x) \leq -1 \).
### Conclusion:
The cosine function, \( \cos(x) \), never reaches values between -1 and 1 (excluding these boundaries). Hence, the secant function, \( \sec(x) \), never takes on values between -1 and 1. Thus, the range of \( y = \sec(x) \) is:
\[ y \in (-\infty, -1] \cup [1, \infty) \]
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Transcribed Image Text:The text in the image reads:
**What is the range of \( y = \sec(x) \)?**
Here is a detailed explanation for an educational website:
---
To understand the range of the secant function, \( y = \sec(x) \), let's first recall that the secant function is the reciprocal of the cosine function, \( \sec(x) = \frac{1}{\cos(x)} \).
### Key Points to Consider:
- The cosine function, \( \cos(x) \), oscillates between -1 and 1 for all \( x \).
- The secant function is undefined where \( \cos(x) = 0 \), which happens at \( x = \frac{\pi}{2} + n\pi \) for any integer \( n \).
### Analyzing the \( \sec(x) \) Function:
- When \( \cos(x) \) is positive and \( 0 < |\cos(x)| \leq 1 \), then \( \sec(x) \geq 1 \).
- When \( \cos(x) \) is negative and \( -1 \leq \cos(x) < 0 \), then \( \sec(x) \leq -1 \).
### Conclusion:
The cosine function, \( \cos(x) \), never reaches values between -1 and 1 (excluding these boundaries). Hence, the secant function, \( \sec(x) \), never takes on values between -1 and 1. Thus, the range of \( y = \sec(x) \) is:
\[ y \in (-\infty, -1] \cup [1, \infty) \]
---
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