Question

Solve the given equation over the interval [0, 2π): csc^2 x + 2 csc x = 0.

Answer 1

`csc^2x+2cscx=0`
Factor out the GCF which is csc x

`cscx(cscx+2)=0`
Set each factor equal to zero.

`cscx=0` or
`cscx+2=0`
Using the reciprocal sine function we can replace csc in the equations.

`(1)/(sinx) =0` or
`(1)/(sinx) +2=0`
The first equation yields 1 = 0 which is not true. However the second equation does offer 2 possible answers.
When solved for sinx you get
`sinx=- (1)/(2)`
The sine function is negative in the 3rd and 4th quadrants.
You can use the unit circle or the
`sin^-^1` feature on your calculator
`sin^-^1(1/2)`

`210^o or 330^o` alter radian answer which is probably what you want is
`(7 \pi )/(6) or (11 \pi )/(6)`