Question

solve for the exact solutions in the interval $0,2\pi)$. if the equation has no solutions, respond with dne.
$3\cot^{2}(x)=7\csc(x)-7$

Explanation:

Step1: Use Pythagorean identity

Recall that \(\cot^{2}x=\csc^{2}x - 1\). Substitute this into the given equation \(3\cot^{2}(x)=7\csc(x)-7\).
We get \(3(\csc^{2}x - 1)=7\csc(x)-7\).
Expand the left - hand side: \(3\csc^{2}x-3 = 7\csc(x)-7\).

Step2: Rearrange the equation

Move all terms to one side to form a quadratic equation. Let \(u = \csc(x)\).
The equation becomes \(3u^{2}-7u + 4=0\).

Step3: Factor the quadratic equation

Factor \(3u^{2}-7u + 4\). We need two numbers \(a\) and \(b\) such that \(a\times b=3\times4 = 12\) and \(a + b=-7\). The numbers are \(-3\) and \(-4\).
\(3u^{2}-3u-4u + 4=0\), then \(3u(u - 1)-4(u - 1)=0\), so \((3u - 4)(u - 1)=0\).

Step4: Solve for \(u\)

Set each factor equal to zero:

• If \(3u-4=0\), then \(u=\csc(x)=\frac{4}{3}\), so \(\sin(x)=\frac{3}{4}\). Using the inverse - sine function, \(x=\sin^{-1}(\frac{3}{4})\) or \(x=\pi-\sin^{-1}(\frac{3}{4})\).
• If \(u - 1=0\), then \(u=\csc(x)=1\), so \(\sin(x)=1\), and \(x=\frac{\pi}{2}\).

Answer:

\(x=\frac{\pi}{2},\sin^{-1}(\frac{3}{4}),\pi-\sin^{-1}(\frac{3}{4})\)