Find all exact radian solutions within the interval [0,2pi). Csc(2x)=-2

Question

Express each final solution as a fraction of the lowest term.

Can someone explain me what procedure do I need to follow for this one?

Answer 1

csc(2x) = 1/sin(2x) = -2

Solve for sin(2x):

sin(2x) = -1/2

Since this is negative, 2x will be in the 3rd and 4th quadrants.

Need to find all angles of sin that have a value of -1/2. For example sin(7π/6) = -1/2.

So 2x = 7π/6 --> x = 7π/12. Look at the trigonometry wheel to find the rest of the angles with sin(θ) = -1/2 and then divide that radian angle by 2.

Maybe they want the negative angles too, sin(-π/6) = -1/2