Question

Which trigonometric functions are equivalent to tan θ? Select all that apply.
(1 point)
Responses

negative tangent left parenthesis negative theta right parenthesis
Image with alt text: negative tangent left parenthesis negative theta right parenthesis

negative tangent theta
Image with alt text: negative tangent theta

tangent left parenthesis negative theta right parenthesis
Image with alt text: tangent left parenthesis negative theta right parenthesis

tangent left-parenthesis theta plus 2 pi right-parenthesis
Image with alt text: tangent left-parenthesis theta plus 2 pi right-parenthesis

tangent left parenthesis theta plus Start Fraction pi over 2 End Fraction right parenthesis

Answer 1

To determine which of the provided options are equivalent to \( \tan \theta \), let’s analyze each one based on trigonometric identities.

1. **\( \tan(-\theta) \)**:
- Using the identity \( \tan(-x) = -\tan(x) \), we have:
\[
\tan(-\theta) = -\tan(\theta)
\]
- Therefore, this is **not equivalent** to \( \tan \theta \).

2. **\( -\tan \theta \)**:
- By definition, this is simply the negative of the tangent function.
- Therefore, this is **not equivalent** to \( \tan \theta \).

3. **\( \tan(-\theta) \)**:
- As mentioned previously, \( \tan(-\theta) = -\tan(\theta) \).
- Therefore, this is **not equivalent** to \( \tan \theta \).

4. **\( \tan(\theta + 2\pi) \)**:
- Using the periodicity of the tangent function, we have:
\[
\tan(\theta + 2\pi) = \tan(\theta)
\]
- Therefore, this is **equivalent** to \( \tan \theta \).

5. **\( \tan(\theta + \frac{\pi}{2}) \)**:
- The tangent function has a vertical asymptote at \( \frac{\pi}{2} + n\pi \) where \( n \) is an integer, meaning \( \tan(\theta + \frac{\pi}{2}) \) is undefined.
- Therefore, this is **not equivalent** to \( \tan \theta \).

Based on the analysis, the only option that is equivalent to \( \tan \theta \) is:

- \( \tan(\theta + 2\pi) \)