`cosec(theta) = -5/3`

But we know, `sin(theta) = 1/(cosec(theta))`

Therefore, `sin(theta) = 1/(-5/3) = -3/5.`

`sin(theta) = -3/5`

Therefore `theta = sin^(-1)(-3/5) = -36.87` degrees.

But we know theta is in Q3, therefore we have to find the general solution for sine in Q3. The general solution for sine is given by,

`theta = n180+(-1)^n(-36.87)` where n is any integer.

n =1 would give the required answer,

`theta = 180+(-1)(-36.87)`

`theta = 180+36.87 = 216.87` degrees

Therefore `2theta = (2)(216.87)` degrees

                       ` = 433.74` degrees

To find the qudrant, we can convert this to a basic angle by reducing 360 from it.

`2theta = 433.74 -360 = 73.74 ` degrees.

Therefore 2theta is in the first quadrant (Q1).

To find `cos(2theta)` we can us the following identity,

`cos(2theta) = 1 - 2sin^(2)(theta)`

`cos(2theta) = 1 - 2(-3/5)^2`

                  `= 0.28`

` cos(2theta) = 0.28`