Question

#50, find the solutions of the equation in the interval [-2pi, 2pi]. Use a graphing utility to verify your results.

Answer 1

Question 50.

Given:

cot x = 1

Let's find the solutions of the given equation in the interval: [-2π, 2π].

To solve for the solutions, apply the following steps:

• Step 1.

Take the inverse cotangent of both sides:

`\begin{gathered} x=\cot ^(-1)(1) \\ \\ x=(\pi)/(4) \end{gathered}`

This function is positive in the first and third quadrants.

• Step 2.

To find the next solution add π to the first solution:

`\begin{gathered} x=(\pi)/(4)+\pi \\ \\ x=(\pi+4\pi)/(4) \\ \\ x=(5\pi)/(4) \end{gathered}`

• Step 3.

Find the period of cotx:

`(\pi)/(|b|)=(\pi)/(|1|)=(\pi)/(1)=\pi`

The period of the cot function will be:

`x=(\pi)/(4)+\pi n,\text{ for any value of n.}`

Substitute -2 for n and solve:

`\begin{gathered} x=(\pi)/(4)+(-2\pi) \\ \\ x=(\pi)/(4)-2\pi \\ \\ x=(\pi-8\pi)/(4) \\ \\ x=-(7\pi)/(4) \end{gathered}`

Substitute -1 for n:

`\begin{gathered} x=(\pi)/(4)+\pi n \\ \\ x=(\pi)/(4)+(-1\pi) \\ \\ x=(\pi)/(4)-\pi \\ \\ x=(\pi-4\pi)/(4) \\ \\ x=-(3\pi)/(4) \end{gathered}`

Substitute 0 for n:

`\begin{gathered} x=(\pi)/(4)+\pi n \\ \\ x=(\pi)/(4)+0 \\ \\ x=(\pi)/(4) \end{gathered}`

Substitute 1 for n:

`\begin{gathered} x=(\pi)/(4)+\pi(1) \\ \\ x=(5)/(4) \end{gathered}`

Therefore, the solutions of the equation in the interval are:

`x=-(7\pi)/(4),-(3\pi)/(4),(\pi)/(4),(5\pi)/(4)`