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

Answer 1

To determine which functions have no zeroes, let's evaluate each option:

1. **y = cos x**: This function has zeroes at odd multiples of π/2 (e.g., π/2, 3π/2, ...). Therefore, it does have zeroes.

2. **y = cot x**: This function has zeroes at integer multiples of π, where x = nπ (n is an integer). Therefore, it does have zeroes.

3. **y = csc x**: The cosecant function is defined as 1/sin x, and since sin x has zeroes at integer multiples of π (where sin x = 0), csc x will be undefined at these points. Therefore, it does not have zeroes.

4. **y = sec x**: The secant function is defined as 1/cos x, and since cos x has zeroes at odd multiples of π/2, sec x will be undefined at these points. Therefore, it does not have zeroes.

5. **y = sin x**: This function has zeroes at integer multiples of π (e.g., 0, π, 2π, ...). Therefore, it does have zeroes.

6. **y = tan x**: This function has zeroes at integer multiples of π where sin x = 0 (x = nπ). Therefore, it does have zeroes.

Based on this analysis, the functions that have no zeroes are:

- **y = csc x**
- **y = sec x**

So the correct answers are **y = csc x** and **y = sec x**.