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

Since angle cita in the third quadrant, then

`\cot \theta=(1)/(2)`

Since tan is the reciprocal of the cot, then

`\begin{gathered} \tan \theta=(1)/(\cot\theta) \\ \tan \theta=(1)/((1)/(2)) \\ \tan \theta=2 \end{gathered}`