leni2uni
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# Solve the given equation over the interval [0, 2π): csc^2 x + 2 csc x = 0.

$csc^2x+2cscx=0$ 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. $\frac{1}{sinx} =0$  or $\frac{1}{sinx} +2=0$ 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 $sinx=- \frac{1}{2}$ 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 $\frac{7 \pi }{6} or \frac{11 \pi }{6}$