Question

Given that csc(0)= and 0 is in Quadrant III, what is tan(0)?Provide your answer below:tan (0) =

Answer 1

Since angle cita lies on the third quadrant, then

$\begin{gathered} \tan \theta>0,\cot \theta>0 \\ \sin \theta<0,\csc \theta<0 \\ \cos \theta<0,\sec \theta<0 \end{gathered}$

From the identity

$\cot ^2\theta=\csc ^2\theta-1$

Substitute csc cita by the given value in the identity

$\begin{gathered} \cot ^2\theta=(-\frac{\sqrt[]{5}}{2})^2-1 \\ \cot ^2\theta=\frac{5}{4}-1 \\ \cot ^2\theta=\frac{1}{4} \\ \cot \theta=\pm\sqrt[]{\frac{1}{4}} \\ \cot \theta=\pm\frac{1}{2} \end{gathered}$

Since angle cita in the third quadrant, then

$\cot \theta=\frac{1}{2}$

Since tan is the reciprocal of the cot, then

$\begin{gathered} \tan \theta=\frac{1}{\cot\theta} \\ \tan \theta=\frac{1}{\frac{1}{2}} \\ \tan \theta=2 \end{gathered}$