Question

Which function has no zeroes? Select all that apply.
(1 point)
Responses

y = cos x
y = cos x

y = cot x
y = cot x

y = csc x
y = csc x

y = sec x
y = sec x

y = sin x
y = sin x

y = tan x
y = tan x

Answer 1

To determine which functions have no zeros, let's analyze each function:

1. **\(y = \cos x\)**: This function has zeros at \(x = \frac{\pi}{2} + n\pi\) (where \(n\) is any integer). So, it has zeros.

2. **\(y = \cot x\)**: The cotangent function is \( \frac{\cos x}{\sin x} \) and has zeros at \(x = n\pi\), where \(n\) is any integer. So, it has zeros.

3. **\(y = \csc x\)**: The cosecant function is the reciprocal of sine, \( \frac{1}{\sin x} \), and it has undefined values (asymptotes) when \(\sin x = 0\) (i.e., at \(x = n\pi\)), but it does not have zeros. It is always defined (except at multiples of \(\pi\)) and never equals zero.

4. **\(y = \sec x\)**: The secant function is the reciprocal of cosine, \( \frac{1}{\cos x} \), and is undefined when \(\cos x = 0\) (i.e., at \(x = \frac{\pi}{2} + n\pi\)), but it does not have zeros. It is always defined (except at odd multiples of \(\pi/2\)) and never equals zero.

5. **\(y = \sin x\)**: This function has zeros at \(x = n\pi\), where \(n\) is any integer. So, it has zeros.

6. **\(y = \tan x\)**: The tangent function is \(\frac{\sin x}{\cos x}\) and has zeros at \(x = n\pi\), where \(n\) is any integer. So, it has zeros.

**Summary**: The functions that have no zeros are:

- \(y = \csc x\)
- \(y = \sec x\)

Thus, the correct responses are:

- \(y = \csc x\)
- \(y = \sec x\)