Find-the-range-of-x-such-that-sinx-gt-0-3-sinx-cosx-gt-0-0-lt-x-lt-2pi-

Question Number 181521 by CrispyXYZ last updated on 26/Nov/22

Find the range of x such that

{\begin{cases} \sin x > 0 \\ \sqrt{3} \sin x + \cos x > 0 \\ 0 < x < 2 π \end{cases}

Answered by mr W last updated on 26/Nov/22

\sin x > 0 \Rightarrow 0 < x < π

\sqrt{3} \sin x + \cos x > 0 \Rightarrow \cot x > - \sqrt{3}

\Rightarrow 0 < x < \frac{5 π}{6}

Answered by mahdipoor last updated on 26/Nov/22

\sqrt{3} s i n x + c o s x = 2 (\frac{\sqrt{3}}{2} s i n x + \frac{1}{2} c o s x) =

2 (c o s 30 s i n x + s i n 30 c o s x) = 2 s i n (30 + x) > 0

\Rightarrow 360 n < 30 + x < 360 n + 180

\Rightarrow s i n x > 0 \Rightarrow 360 n < x < 360 n + 180

\Rightarrow 0 < x < 360

\Rightarrow\Rightarrow\Rightarrow 0 < x < 150 o r 0 < x < \frac{5}{6} π