For this problem, we are given a certain equation, and we need to solve it over the interval [0º, 360º).
The equation is:
[tex]-8\csc ^2(x)+2\cot (x)=-23_{}[/tex]We can represent the csc as a cot, as shown below:
[tex]\begin{gathered} -8(1+\cot ^2(x))+2\cot (x)=-23_{} \\ -8-8\cot ^2(x)+2\cot (x)=-23 \\ -8\cot ^2(x)+2\cot (x)=-23+8 \\ -8\cot ^2(x)+2\cot (x)=-15 \\ -8\cot ^2(x)+2\cot (x)+15=0 \end{gathered}[/tex]Now we can use an auxiliary variable, "y", such as:
[tex]y=\cot (x)[/tex]And we can rewrite the expression as:
[tex]-8y^2+2y+15=0[/tex]We can solve for "y", such as:
[tex]\begin{gathered} y_{1,2}=\frac{-2\pm\sqrt[]{(2)^2-4\cdot(-8)\cdot(15)}}{2\cdot(-8)} \\ y_{1,2}=\frac{-2\pm\sqrt[]{484}}{-16} \\ y_{1,2}=\frac{-2\pm22}{-16} \\ y_1=\frac{-2-22}{-16}=1.5 \\ y_2=\frac{-2+22}{-16}=-1.25 \end{gathered}[/tex]We have two possible values for "y", and we need to solve the auxiliary equation to determine the solutions:
[tex]\begin{gathered} \cot (x)=1.5 \\ x=\text{arccot}(1.5) \\ x=3.1416\cdot n+0.588 \end{gathered}[/tex][tex]\begin{gathered} \cot (x)=-1.25 \\ x=\text{arccot}(-1.25) \\ x=3.1416\cdot n-0.67474 \end{gathered}[/tex]Since the cosecant is odd, we also have:
[tex]\begin{gathered} x=3.1416\cdot n-2.55 \\ x=3.1416\cdot n+2.466 \end{gathered}[/tex]Since we want the values for x between 0 and 360º, which would be the same as 0 and 2pi rad, we have:
[tex]\begin{gathered} x=0.588\text{ rad}=33.69º \\ x=3.7296\text{ rad = 213.69º} \\ x=2.4668\text{ rad}=141.3º \\ x=5.6076\text{ rad}=321.29º \end{gathered}[/tex]The solutions are: 33.69º, 213.69º, 141.3º and 321.29º
