1-cos-x-3-sin-x-4-

Question Number 106295 by bobhans last updated on 04/Aug/20

\frac{1}{\cos x} + \frac{\sqrt{3}}{\sin x} = 4

Answered by bemath last updated on 04/Aug/20

\sin x + \sqrt{3} \cos x = 4 \sin xcos x

\frac{1}{2} \sin x + \frac{\sqrt{3}}{2} \cos x = \sin 2 x

\sin 30 ° \sin x + \cos 30 ° \cos x = \sin 2 x

\cos (30 ° - x) = \sin 2 x = \cos (90 ° - 2 x)

{\begin{cases} 30 ° - x = 90 ° - 2 x + k . 360 ° \\ 30 ° - x = - 90 ° + 2 x + k . 360 ° \end{cases}

{\begin{cases} x = 60 ° + k . 360 ° \\ x = 40 ° + k . 120 ° \end{cases}

Answered by Dwaipayan Shikari last updated on 04/Aug/20

s i n x + \sqrt{3} c o s x = 4 s i n x c o s x

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

s i n (x + \frac{π}{3}) = s i n 2 x

2 x = 2 π k \pm (x + \frac{π}{3})

x = 2 π k + \frac{π}{3} = (6 k + 1) \frac{π}{3} (k \in Z)

3 x = 2 π k - \frac{π}{3}

x = (6 k - 1) \frac{π}{9}