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.

choose All that appply

Answer 1

To determine which of the provided functions are equivalent to \(\tan \theta\), we will analyze each option based on trigonometric identities:

1. **\(-\tan(-\theta)\)**:
- Using the identity \(\tan(-\theta) = -\tan(\theta)\), we have:
\[
-\tan(-\theta) = -(-\tan(\theta)) = \tan(\theta)
\]
This is equivalent to \(\tan \theta\).

2. **\(-\tan \theta\)**:
- This is simply the negative of \(\tan \theta\) and is **not** equivalent to \(\tan \theta\).

3. **\(\tan(-\theta)\)**:
- By the identity \(\tan(-\theta) = -\tan(\theta)\), this is **not** equivalent to \(\tan \theta\).

4. **\(\tan(\theta + 2\pi)\)**:
- The tangent function is periodic with a period of \(\pi\), so:
\[
\tan(\theta + 2\pi) = \tan(\theta)
\]
This is equivalent to \(\tan \theta\).

5. **\(\tan\left(\theta + \frac{\pi}{2}\right)\)**:
- The tangent function has a vertical asymptote at \(\theta = \frac{\pi}{2}\) plus multiples of \(\pi\), thus:
\[
\tan\left(\theta + \frac{\pi}{2}\right) = -\cot(\theta)
\]
This is **not** equivalent to \(\tan\theta\).

Based on this analysis, the functions that are equivalent to \(\tan \theta\) are:

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

### Final Answer:
- \(-\tan(-\theta)\)
- \(\tan(\theta + 2\pi)\)