Question

In which quadrant does θ lie if the following statements are true: csc θ> 0 and sec θ> 0

Answer 1

ANSWER

First quadrant

EXPLANATION

Those two trigonometric functions are the reciprocal of the sine and cosine functions:

$\begin{gathered} \sec \theta=\frac{1}{\cos \theta} \\ \csc \theta=\frac{1}{\sin \theta} \end{gathered}$

Therefore:

$\csc \theta>0\Rightarrow\sin \theta>0$

sinθ is greater than zero for the first and second quadrants.

Also:

$\sec \theta>0\Rightarrow\cos \theta>0$

cosθ is greater than zero for the first and fourth quadrants.

Hence, for cscθ>0 and secθ>0, angle θ lies in the first quadrant.