Question

\[ \tan \left( -\frac{23}{3} \pi \right) = \]

• A. \( -\frac{1}{\sqrt{3}} \)
• B. \( \frac{1}{\sqrt{3}} \)
• C. \( -\sqrt{3} \)
• D. \( \sqrt{3} \)

Hint:

For tangent functions with angles involving multiples of \( \pi \), use the periodicity of the tangent function to simplify the argument.

Answer 1

The Correct Option is D

We know that \( \tan(\theta + n\pi) = \tan(\theta) \), where \( n \) is an integer. First, simplify the argument of the tangent: \[ -\frac{23}{3} \pi = -8\pi - \frac{\pi}{3}. \] Since \( \tan(\theta + \pi) = -\tan(\theta) \), this becomes: \[ \tan\left(-8\pi - \frac{\pi}{3}\right) = \tan\left(\frac{\pi}{3}\right) = \sqrt{3}. \]