choose an appropriate viewing window to display the trigonometric function in radian mode. show all written work for this problem
a.)y = sec x b.) y= csc x c.) y=cot x

To choose an appropriate viewing window for displaying trigonometric functions, we need to understand the properties of each function, particularly the period, asymptotes, and general behavior of the function.

### a) \( y = \sec(x) \)
The secant function is defined as \( \sec(x) = \frac{1}{\cos(x)} \). It is undefined where \( \cos(x) = 0 \), which occurs at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer. The function has a period of \( 2\pi \).

- **Period:** \( 2\pi \)
- **Asymptotes:** \( x = \frac{\pi}{2} + n\pi \)
- **Range:** \( (-\infty, -1] \cup [1, \infty) \)

**Appropriate Viewing Window:**
- \( x \)-interval: \([0, 2\pi]\)
- \( y \)-interval: \([-3, 3]\)

### b) \( y = \csc(x) \)
The cosecant function is defined as \( \csc(x) = \frac{1}{\sin(x)} \). It is undefined where \( \sin(x) = 0 \), which occurs at \( x = n\pi \), where \( n \) is an integer. The function has a period of \( 2\pi \).

- **Period:** \( 2\pi \)
- **Asymptotes:** \( x = n\pi \)
- **Range:** \( (-\infty, -1] \cup [1, \infty) \)

**Appropriate Viewing Window:**
- \( x \)-interval: \([0, 2\pi]\)
- \( y \)-interval: \([-3, 3]\)

### c) \( y = \cot(x) \)
The cotangent function is defined as \( \cot(x) = \frac{1}{\tan(x)} \). It is undefined where \( \tan(x) = 0 \), which occurs at \( x = n\pi \), where \( n \) is an integer. The function has a period of \( \pi \).

- **Period:** \( \pi \)
- **Asymptotes:** \( x = n\pi \)
- **Range:** \( (-\infty, \infty) \)

**Appropriate Viewing Window:**
- \( x \)-interval: \([0, \pi]\)
- \( y \)-interval: \([-3, 3]\)

### Summary of Viewing Windows:
- \( y = \sec(x) \): \( x \in [0, 2\pi] \), \( y \in [-3, 3] \)
- \( y = \csc(x) \): \( x \in [0, 2\pi] \), \( y \in [-3, 3] \)
- \( y = \cot(x) \): \( x \in [0, \pi] \), \( y \in [-3, 3] \)

This setup ensures that the functions are fully displayed with their key features (like asymptotes and periodic behavior) visible within the window.

Would you like more details on any of these functions or another example?

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Here are five questions to deepen your understanding:
1. How do the asymptotes of \( y = \sec(x) \) and \( y = \csc(x) \) influence the graph?
2. What happens to the \( y \)-values of \( y = \sec(x) \) and \( y = \csc(x) \) near their asymptotes?
3. Can you describe the general shape of the \( y = \cot(x) \) graph within one period?
4. Why does \( y = \sec(x) \) have the same period as \( y = \csc(x) \), even though they are based on different trigonometric functions?
5. How would the graph of \( y = \sec(x) \) change if the equation was \( y = 2\sec(x) \)?

**Tip:** Understanding the periodicity and asymptotes of trigonometric functions is crucial for correctly setting up viewing windows and interpreting their graphs.

choose an appropriate viewing window to display the trigonometric function in radian mode. show all written work for this problem
a.) y = sec x
b.) y = csc x
c.) y = cot x

Explore how to choose an appropriate viewing window for the trigonometric functions y = sec(x), y = csc(x), and y = cot(x) in radian mode. This guide includes key trigonometric concepts, formulas, and theorems relevant to high school math students in Grades 10-12.

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