Question

If $0\le x\le 2\pi$ and $\sin x<0$, find the correct inequality among the following:
I. $\cos x<0$
II. $cosecx<0$
III. $|\sin x+\cos x|>0$

• A. I only
• B. II only
• C. III only
• D. I and II
• E. II and III

Answer 1

The correct option is E II and III

$0\le x\le 2\pi ,\sin x<0$

$\sin x<0$

$\Rightarrow \pi <x<2\pi$

$Between\pi <x<2\pi$

$\cos x\ge 0,\frac{3\pi }{2}<x<2\pi$

$\cos x\le 0,\pi <x\frac{3\pi }{2}$

$\csc x<1,\pi <x<2\pi$

$\sin x+\cos x$ is negative and positive for different values between $[\pi ,2\pi ]$

Only II is true.