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

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[latex]csc^2x+2cscx=0[/latex] Factor out the GCF which is csc x [latex]cscx(cscx+2)=0[/latex] Set each factor equal to zero. [latex]cscx=0[/latex]  or [latex]cscx+2=0[/latex] Using the reciprocal sine function we can replace csc in the equations. [latex] \frac{1}{sinx} =0[/latex]  or [latex] \frac{1}{sinx} +2=0[/latex] 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 [latex]sinx=- \frac{1}{2} [/latex] The sine function is negative in the 3rd and 4th quadrants. You can use the unit circle or the [latex]sin^-^1[/latex] feature on your calculator  [latex]sin^-^1(1/2)[/latex] [latex]210^o or 330^o[/latex]  alter radian answer which is probably what you want is [latex] \frac{7 \pi }{6} or \frac{11 \pi }{6} [/latex]

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