find all solutions to the equation √3 csc(2theta)=-2

Question

Would the answer be
π/6 + 2πn, 5π/6 +2πn or π/6 + πn, 5π/6 +πn?

or neither?

Answer 1

sqrt ( 3 ) * csc ( 2 theta ) = - 2 Divide both sides by sqrt ( 3 )

csc ( 2 theta ) = - 2 / sqrt ( 3 )

Take the inverse cosecant of both sides.

2 theta = - pi / 3

and

2 theta = 4 pi / 3

[ Becouse csc ( pi / 3 ) = 2 / sqrt ( 3 ) , and csc ( 4 pi / 3 = 2 / sqrt ( 3 ) ]

Divide both sides by 2

theta = - pi / 6

and

theta = 2 pi / 3

Period of csc ( theta ) = 2 pi

So period of 2 theta = 2 pi / 2 = pi

Final solutions :

theta = n pi + 2 pi / 3 = pi ( n + 2 / 3 )

and

theta = n pi - pi / 6 = pi ( n - 1 / 6 )