All possible values of [0,2] for which 2+20 lie in:

Question

All possible values of $	heta\in[0,2\pi]$ for which $\sin2	heta+	an2	heta>0$ lie in:

Accepted Answer

$$\sin2\theta+\tan2\theta = \tan2\theta(1+\cos2\theta) > 0$$

Since $1+\cos2\theta \ge 0$ always (and $=0$ only where $\cos2\theta=-1$, excluded), the condition reduces to $\tan2\theta>0$ (with $\cos2\theta\ne0$, i.e. $\sin\theta\ne\pm\tfrac1{\sqrt2}$):

$$2\theta \in \left(0,\tfrac\pi2\right)\cup\left(\pi,\tfrac{3\pi}2\right)\cup\left(2\pi,\tfrac{5\pi}2\right)\cup\left(3\pi,\tfrac{7\pi}2\right)$$

$$\theta \in \left(0,\tfrac\pi4\right)\cup\left(\tfrac\pi2,\tfrac{3\pi}4\right)\cup\left(\pi,\tfrac{5\pi}4\right)\cup\left(\tfrac{3\pi}2,\tfrac{7\pi}4\right)$$

Solutions · 1