Question

Which of the following functions have a vertical asymptote for values of θ such that cos θ = 1? Select all that apply.
(1 point)
Responses

y = sin θ
y = sin θ

y = cos θ
y = cos θ

y = tan θ
y = tan θ

y = sec θ
y = sec θ

y = csc θ
y = csc θ

y = cot θ

Answer 1

To determine which functions have a vertical asymptote for values of \( \theta \) such that \( \cos \theta = 1 \), we need to consider what happens at those values.

The values of \( \theta \) where \( \cos \theta = 1 \) occur at:
\[
\theta = 2n\pi, \quad n \in \mathbb{Z}
\]
This corresponds to the angles where the cosine function reaches its maximum value of 1.

Now let's analyze each function for vertical asymptotes:

1. **\( y = \sin \theta \)**: The sine function does not have vertical asymptotes. It oscillates between -1 and 1 for all \( \theta \). So, **no vertical asymptote** here.

2. **\( y = \cos \theta \)**: As with the sine function, the cosine function also does not have vertical asymptotes and oscillates between -1 and 1. So, **no vertical asymptote** here.

3. **\( y = \tan \theta \)**: The tangent function has vertical asymptotes where \( \cos \theta = 0 \), not where \( \cos \theta = 1 \). So, **no vertical asymptote** here.

4. **\( y = \sec \theta \)**: The secant function is defined as \( \sec \theta = \frac{1}{\cos \theta} \). When \( \cos \theta = 1 \) (e.g., at \( \theta = 0, 2\pi, \ldots \)), the secant is defined and equals 1, so there are **no vertical asymptotes** for this function.

5. **\( y = \csc \theta \)**: The cosecant function is defined as \( \csc \theta = \frac{1}{\sin \theta} \). While this function has vertical asymptotes at \( \theta = n\pi \) (where \( \sin \theta = 0 \)), it does not have vertical asymptotes at points where \( \cos \theta = 1 \), so **no vertical asymptote** here.

6. **\( y = \cot \theta \)**: The cotangent function is \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). It has vertical asymptotes where \( \sin \theta = 0 \). Since these asymptotes do not occur at points where \( \cos \theta = 1 \), there are **no vertical asymptotes** for this function either.

In conclusion, none of the provided functions have vertical asymptotes for values of \( \theta \) where \( \cos \theta = 1 \).

**Answer:** None of the functions have vertical asymptotes where \( \cos \theta = 1 \).