Question

answer:

8 $y = csc x cot x, y(\frac{pi}{2}) = 6$

to advance in the circuit, locate $y(\frac{11pi}{6})$.

Explanation:

Step1: Integrate \(y'\)

We know that \(\int\csc x\cot xdx =-\csc x + C\). So \(y =-\csc x + C\).

Step2: Find \(C\)

Given \(y(\frac{\pi}{2}) = 6\). Substitute \(x=\frac{\pi}{2}\) into \(y =-\csc x + C\). Since \(\csc\frac{\pi}{2}=1\), we have \(6=-1 + C\), then \(C = 7\). So \(y=-\csc x+7\).

Step3: Calculate \(y(\frac{11\pi}{6})\)

Substitute \(x = \frac{11\pi}{6}\) into \(y=-\csc x+7\). Since \(\csc\frac{11\pi}{6}=\csc(2\pi-\frac{\pi}{6})=-\csc\frac{\pi}{6}=-2\), then \(y= -(-2)+7=9\).

Answer:

\(9\)