Which trigonometric functions are equivalent to tan θ? Select all that apply.

Question

(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:

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