Question

In which quadrant will θ lie if csc θ > 0 and cos θ < 0?

Answer 1

$\csc\theta=\dfrac{1}{\sin\theta}$

We know:

$\sin\theta > 0$ in I and II quadrant

$\cos\theta < 0$ in II and III quadrant

Therefore your answer is:

θ is in II quadrant

Answer 2

The angle θ will lie in the second quadrant, that is, quadrant II, when the trigonometric functions csc θ > 0 and cos θ < 0.

How do the values of trigonometric functions depend on the quadrant in which the angle lies?

For the function sin θ:

Quadrant I: sin θ > 0.

Quadrant II: sin θ > 0.

Quadrant III: sin θ < 0

Quadrant IV: sin θ < 0.

For the function cos θ:

Quadrant I: cos θ > 0.

Quadrant II: cos θ < 0.

Quadrant III: cos θ < 0

Quadrant IV: cos θ > 0.

All rest functions can be calculated in terms of sine and cosine functions.

How to solve the question?

In the question, we are asked to tell the quadrant in which θ lies, if csc θ > 0 and cos θ < 0.

We know that csc θ = 1/ sin θ.

Therefore, when csc θ > 0, sin θ > 0, which implies that θ lies in either quadrant I or quadrant II.

But, we are also given that cos θ < 0, which is when θ lies in either quadrant II or quadrant III.

Hence, on analyzing both the functions, we can say that θ lies in quadrant II, when csc θ > 0 and cos θ < 0.

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