Question

Solve 2csc^2x-2cscx-1=0
for 0° ≤ x ≤ 360°

Answer 1

Explanation:

2 csc²x-2 csc x-1=0

or

`(2)/(sin^2x) -(2)/(sin x) -1=0\\multiply~by~sin^2x\\2-2sin~x-sin^2x=0\\sin^2x+2sinx-2=0\\sin ~x=(-2 \pm√(2^2-4*1*(-2)) )/(2*1) \\=(-2 \pm√(4+8) )/(2) \\=(-2 \pm√(4*3) )/(2) \\=(-2 \pm2√(3) )/(2) \\=-1\pm√(3) \\|sin~x| \le ~1\\so~sin~x=-1+√(3) \\sin~x=√(3) -1\\x=sin^(-1)(√(3) -1) \approx 47.06^\circ,180-47.06\\or~x=47.06^\circ,132.94^\circ`