Chapter 5, Problem 31RE

Solution

To calculate: The general solution of the equation $\tan x=-1$ .

The solution is $x=\frac{3\pi }{4}+\pi n$, for any integer $n$ .

Given information:

The equation is $\tan x=-1$ .

Calculation:

Take the inverse tangent on both sides of the given equation.

  $x={\tan }^{-1}\left(-1\right)$

The exact value of ${\tan }^{-1}\left(-1\right)$ is $-\frac{\pi }{4}$ .

Consequently, $x=-\frac{\pi }{4}$ .

Since, the tangent function is negative in the second and fourth quadrants. So, the second solution will be $x=-\frac{\pi }{4}-\pi$ or $x=\frac{3\pi }{4}$ .

Also, the period of tangent function is $\pi$ . So, the value of tangent function will repeat after $\pi$ radians in both the directions.

Therefore, the general solution of the given equation will be $x=\frac{3\pi }{4}+\pi n$, for any integer $n$ .