Question

In Exercises \(31-36,\) solve the equation in the specified interval. $$\cot x=-1\( \)-\infty$$

Short answer

The solution to the equation \(\cot x = -1\) within the interval \(-\infty < x < \infty\) is \(x = 2n\pi + 3\pi/4\), where n is an integer.

Step-by-step solution

Understand the cotangent function

The cotangent function, \(\cot x\), is a periodic function with a period of \(\pi\). It takes the value of -1 at odd multiples of \(\pi/4\) within each interval of \(\pi\). This means \(\cot x = -1\) if \(x = 2n\pi + 3\pi/4\), where n is an integer.

Solve for x within the specified interval

The task is to find values of x within the interval from \(-\infty\) to \(\infty\) that satisfy the equation \(\cot x = -1\). We can rewrite the equation as \(x = 2n\pi + 3\pi/4\), where n is an integer. Setting n to different integer values will give the different solutions that satisfy the equation within the given interval.

Conclude the solution

The solution to the equation \(\cot x = -1\) in the interval from \(-\infty\) to \(\infty\) are all values of x of the form \(x = 2n\pi + 3\pi/4\), where n is an integer.