Discussion
If sin \(\alpha = \frac{12}{13}\), where \(\frac{\pi}{2} \lt \alpha \lt \frac{3\pi}{2}\) then the value of tan \(\alpha\) is equal to
Answer: C
\(\tan\alpha = \) \(\frac{-12}{5}\).
Between \(\frac{\pi}{2}\) and \(\frac{3\pi}{2}\) the sine is positive only in the second quadrant, so \(\alpha\) is there and \(\cos\alpha\) is negative:
\[ \begin{aligned} \cos\alpha &= -\sqrt{1 - \tfrac{144}{169}} = -\tfrac{5}{13} \\ \tan\alpha &= \frac{12/13}{-5/13} = -\frac{12}{5} \end{aligned} \]
\(\frac{5}{12}\) is the cotangent with the sign dropped; \(\frac{13}{5}\) and \(\frac{-13}{5}\) are \(\pm\sec\alpha\), not \(\tan\alpha\).
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