Question

Solve the following equation on the interval [0°, 360º). Round answers to the nearest tenth. If there is no solution, indicate "No Solution."-8csc^2 (x) + 2cot(x) = -23

Answer 1

`x=141.34,33.69`

Explanation:

We have the following equation:

`-8\csc ^2(x)+2\cot \mleft(x\mright)=-23`

We sovle for x:

`\begin{gathered} \csc ^2(x)=1+\cot ^2(x) \\ \text{ We replacing} \\ -8\cdot(1+\cot ^2(x))+2\cot (x)=-23 \\ -8-8\cot ^2(x)+2\cot (x)+23=0 \\ -8\cot ^2(x)+2\cot (x)+15=0 \\ \text{ Using the substitution method:} \\ \cot (x)=u \\ -8u^2+2u+15=0 \\ -1\cdot(8u^2-2u-15)=0 \\ 8u^2-2u-15=0 \\ -2u=+10u-12u \\ 8u^2+10u-12u-15=0 \\ 2u(4u+5)-3u(4u+5)=0 \\ (4u+5)(2u-3)=0 \\ 4u+5=0\rightarrow u=-(5)/(4) \\ 2u+-3=0\rightarrow u=(3)/(2) \\ \text{Therefore:} \\ \cot (x)=-(5)/(4)\rightarrow x=\cot ^(-1)\mleft(-(5)/(4)\mright)\rightarrow x=141.34\degree \\ \cot (x)=(3)/(2)\rightarrow x=\cot ^(-1)\mleft((3)/(2)\mright)\rightarrow x=33.69\degree \end{gathered}`