What is the equation for a cosecant function with vertical asymptotes found at  x equals pi over 2 plus pi over 2 times n comma  such that n is an integer?

The cosecant function \( \csc(x) \) has vertical asymptotes where its argument equals \( k\pi \), where \( k \) is an integer. However, in your case, the vertical asymptotes are given as occurring at \( x = \frac{\pi}{2} + \frac{\pi}{2}n \), where \( n \) is an integer. 

To match this condition, consider the standard cosecant function \( \csc(kx) \), where \( k \) is a constant that affects the frequency of the function. Let's derive the appropriate function:

1. The standard vertical asymptotes of \( \csc(kx) \) occur at \( kx = m\pi \), where \( m \) is an integer.
2. We need \( x = \frac{\pi}{2} + \frac{\pi}{2}n \) to match the asymptotes of this function.
3. Set \( kx = \frac{\pi}{2} + \frac{\pi}{2}n \).

To satisfy the condition:

\[ kx = \frac{\pi}{2} \text{ when } x = \frac{\pi}{2}, \text{ and } kx = \frac{\pi}{2} + \frac{\pi}{2}n \text{ when } x = \frac{\pi}{2} + \frac{\pi}{2}n \]

This implies \( kx = \frac{\pi}{2}(1 + n) \), so \( k = \frac{1 + n}{1 + n} = 1 \). However, because you need the frequency altered so that \( kx = \frac{\pi}{2} + \frac{\pi}{2}n \), we set \( k = 2 \).

Thus, the correct function would be:

\[
y = \csc\left(2x\right)
\]

The vertical asymptotes of \( y = \csc(2x) \) will occur where \( 2x = (2n+1)\frac{\pi}{2} \), meaning \( x = \frac{\pi}{2} + \frac{\pi}{2}n \), which matches the condition provided.

### Summary
The equation for the cosecant function with vertical asymptotes at \( x = \frac{\pi}{2} + \frac{\pi}{2}n \) is:

\[
y = \csc(2x)
\]

Would you like more details, or do you have any questions?

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**Related Questions:**
1. What are the general properties of the cosecant function?
2. How do phase shifts affect the position of vertical asymptotes in trigonometric functions?
3. How does the period of a trigonometric function change with a coefficient inside the function?
4. Can you explain how vertical asymptotes are determined for other trigonometric functions like secant or tangent?
5. What would be the equation if the vertical asymptotes were at \( x = \frac{\pi}{4} + \frac{\pi}{4}n \)?

**Tip:** Understanding how the coefficient inside the argument of trigonometric functions affects their period and asymptotes is crucial for graphing these functions accurately.

Learn the equation for a cosecant function with vertical asymptotes at x = pi/2 + pi/2*n, involving trigonometric concepts and specific solutions. Explore how trigonometric functions like cosecant are affected by vertical asymptotes and their implications.

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