Question

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

Answer 1

For this problem, we are given a certain equation, and we need to solve it over the interval [0º, 360º).

The equation is:

`-8\csc ^2(x)+2\cot (x)=-23_{}`

We can represent the csc as a cot, as shown below:

`\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}`

Now we can use an auxiliary variable, "y", such as:

`y=\cot (x)`

And we can rewrite the expression as:

`-8y^2+2y+15=0`

We can solve for "y", such as:

`\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)=(-2\pm22)/(-16) \\ y_1=(-2-22)/(-16)=1.5 \\ y_2=(-2+22)/(-16)=-1.25 \end{gathered}`

We have two possible values for "y", and we need to solve the auxiliary equation to determine the solutions:

`\begin{gathered} \cot (x)=1.5 \\ x=\text{arccot}(1.5) \\ x=3.1416\cdot n+0.588 \end{gathered}`
`\begin{gathered} \cot (x)=-1.25 \\ x=\text{arccot}(-1.25) \\ x=3.1416\cdot n-0.67474 \end{gathered}`

Since the cosecant is odd, we also have:

`\begin{gathered} x=3.1416\cdot n-2.55 \\ x=3.1416\cdot n+2.466 \end{gathered}`

Since we want the values for x between 0 and 360º, which would be the same as 0 and 2pi rad, we have:

`\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}`

The solutions are: 33.69º, 213.69º, 141.3º and 321.29º