Question

Determine the quadrant when the terminal side of the angle lies according to the following conditions: $\tan \left(t\right)>0,cosec\left(t\right)<0$

• A. Quadrant $I$
• B. Quadrant $III$
• C. Quadrant $IV$
• D. Quadrant $II$

Answer 1

The correct option is B

Quadrant $III$

Trigonometry functions in all quadrant:

In the first quadrant the trigonometric function $\sin \left(t\right),\cos \left(t\right),\tan \left(t\right),cosec\left(t\right),\sec \left(t\right),\cot \left(t\right)$ all are positive.

In the second quadrant trigonometric function $\sin \left(t\right),cosec\left(t\right)$ are positive.

In the third quadrant trigonometric function $\tan \left(t\right),\cot \left(t\right)$ are positive.

In the fourth quadrant trigonometric function $\cos \left(t\right),\sec \left(t\right)$ are positive.

In the first quadrant $\tan \left(t\right)>0$ but $cosec\left(t\right)<0$ is not possible.

Thus, first option is discarded.

Thus, $cosec\left(t\right)<0$ in third and fourth quadrant and $\tan \left(t\right)>0$ in the third quadrant.

Thus, second option is correct.

In the fourth quadrant $\tan \left(t\right)>0$ but $cosec\left(t\right)<0$ are not possible.

Thus, third option is discarded.

In the second quadrant $\tan \left(t\right)>0$ and $cosec\left(t\right)<0$ are not possible.

Hence, fourth option is discarded.

Therefore, option B is the correct answer.