Question

In which quadrant does θ lie if the following statements are true:
cosθ<0 and cscθ>0

Answer 1

Given:

`\cos\theta <0` and
`\csc\theta >0`

To find:

The quadrant in which θ lie if the given statements are true.

Solution:

We know that,

All trigonometric ratios are positive in I quadrant.

Only sine and cosecant are positive in II quadrant and others are negative.

Only tangent and cotangent are positive in III quadrant and others are negative.

Only cosine and secant are positive in IV quadrant and others are negative.

We have,

`\cos\theta <0` and
`\csc\theta >0`

Here, cosine is negative and cosecant is positive. It is only possible when
`\theta` lies in II quadrant.

Therefore, the
`\theta` lies in II quadrant.