Question

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

Answer 1

First quadrant

Step-by-step explanation

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

`\begin{gathered} \sec \theta=(1)/(\cos \theta) \\ \csc \theta=(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.