Question

what is cot \frac{11\pi}{4}?

Explanation:

Step1: Use the cotangent periodicity

The cotangent function has a period of \(\pi\), so \(\cot(x + n\pi)=\cot(x)\) for any integer \(n\).
\(\frac{11\pi}{4}=2\pi+\frac{3\pi}{4}\), then \(\cot(\frac{11\pi}{4})=\cot(2\pi + \frac{3\pi}{4})\)
Since \(n = 2\) (an integer), \(\cot(2\pi+\frac{3\pi}{4})=\cot(\frac{3\pi}{4})\)

Step2: Use the co - function identity

We know that \(\cot(x)=\frac{\cos(x)}{\sin(x)}\), and for \(x=\frac{3\pi}{4}\), \(\cos(\frac{3\pi}{4})=-\frac{\sqrt{2}}{2}\), \(\sin(\frac{3\pi}{4})=\frac{\sqrt{2}}{2}\)
\(\cot(\frac{3\pi}{4})=\frac{\cos(\frac{3\pi}{4})}{\sin(\frac{3\pi}{4})}=\frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}\)

Step3: Simplify the expression

\(\frac{-\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}=- 1\)

Answer:

\(-1\)