Solve the following equation exactly: csc θ = − 2, 0 ≤ θ < 4π?

cscθ = -2
1/sinθ = -2
sinθ = -1/2
θ = π+(π/6), 2π-(π/6), 2π+π+(π/6), 2π+2π-(π/6)
θ = 7π/6, 11π/6, 19π/6, 23π/6

So basically you want
Csc x = -2
Sin x = -1/2
X = 7pi/6 +2pi *K , 11pi/6 + 2pi* K

sin Ө = - 1 / 2
Ө = 7π/6 , 11 π/6 , 19π/6 , 23π/6

First we want to turn this into a simple trigonometric equation of the form sin(x) = a:
csc(θ) = -2
1/sin(θ) = -2
sin(θ) = -1/2

Then we want to find the elementary angle:
arcsin(-1/2) = -pi/6

Then we want to find the general equation for θ:
θ = 2n*pi - pi/6 or θ = (2n+1)*pi + pi/6 where n E Z

Now we want to find all solutions along the domain θ E [0, 4pi]:
θ = (2pi - pi/6), (4pi - pi/6), (pi + pi/6), (3pi + pi/6)
θ = 11pi/6, 23pi/6, 7pi/6, 19pi/6

Now we want to order these solutions from lowest to highest:
θ = 7pi/6, 11pi/6, 19pi/6, 23pi/6

And that's your answer!

csc θ = 1/(sin θ)
=>
-2 sin θ = 1
sin θ = -1/2
θ = {7pi/6, 11pi/6, 19pi/6, 23pi/6}