All possible values of theta in [0, 2pi] for | vedclass

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Question

(D) Given inequality: $\sin 2	heta + 	an 2	heta > 0$$\Rightarrow \sin 2	heta + \frac{\sin 2	heta}{\cos 2	heta} > 0$$\Rightarrow \sin 2	heta \le

Vedclass · Accepted Answer

$\left(0, \frac{\pi}{4}\right) \cup \left(\frac{\pi}{2}, \frac{3\pi}{4}\right) \cup \left(\pi, \frac{5\pi}{4}\right) \cup \left(\frac{3\pi}{2}, \frac{7\pi}{4}\right)$

Vedclass · Answer

(D) Given inequality: $\sin 2	heta + 	an 2	heta > 0$$\Rightarrow \sin 2	heta + \frac{\sin 2	heta}{\cos 2	heta} > 0$$\Rightarrow \sin 2	heta \left(1 + \frac{1}{\cos 2	heta}ight) > 0$$\Rightarrow \sin 2	heta \left(\frac{\cos 2	heta + 1}{\cos 2	heta}ight) > 0$$\Rightarrow 	an 2	heta (2 \cos^2 	heta) > 0$Since $2 \cos^2 	heta \ge 0$,for the expression to be $> 0$,we must have $	an 2	heta > 0$ and $\cos 2	heta 
eq 0$ (which implies $\cos^2 	heta 
eq 0$).$	an 2	heta > 0$ occurs when $2	heta \in (0, \frac{\pi}{2}) \cup (\pi, \frac{3\pi}{2}) \cup (2\pi, \frac{5\pi}{2}) \cup (3\pi, \frac{7\pi}{2})$.Dividing by $2$,we get $	heta \in (0, \frac{\pi}{4}) \cup (\frac{\pi}{2}, \frac{3\pi}{4}) \cup (\pi, \frac{5\pi}{4}) \cup (\frac{3\pi}{2}, \frac{7\pi}{4})$.Thus,the correct option is $D$.

All possible values of $\theta \in [0, 2\pi]$ for which $\sin 2\theta + \tan 2\theta > 0$ lie in

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