ctg(x-П)<√21 inequation

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cot(x - pi) < \/ 21

\cot{\left(x - \pi \right)} < \sqrt{21}

cot(x - pi) < sqrt(21)

Detail solution

Given the inequality:

\cot{\left(x - \pi \right)} < \sqrt{21}

To solve this inequality, we must first solve the corresponding equation:

\cot{\left(x - \pi \right)} = \sqrt{21}

Solve:
Given the equation

\cot{\left(x - \pi \right)} = \sqrt{21}

- this is the simplest trigonometric equation
This equation is transformed to

x = \pi n + \operatorname{acot}{\left(\sqrt{21} \right)}

Or

x = \pi n + \operatorname{acot}{\left(\sqrt{21} \right)}

, where n - is a integer

x_{1} = \pi n + \operatorname{acot}{\left(\sqrt{21} \right)}

x_{1} = \pi n + \operatorname{acot}{\left(\sqrt{21} \right)}

This roots

x_{1} = \pi n + \operatorname{acot}{\left(\sqrt{21} \right)}

is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:

x_{0} < x_{1}

For example, let's take the point

x_{0} = x_{1} - \frac{1}{10}

=

\left(\pi n + \operatorname{acot}{\left(\sqrt{21} \right)}\right) - \frac{1}{10}

=

\pi n - \frac{1}{10} + \operatorname{acot}{\left(\sqrt{21} \right)}

substitute to the expression

\cot{\left(x - \pi \right)} < \sqrt{21}

\cot{\left(\left(\pi n - \frac{1}{10} + \operatorname{acot}{\left(\sqrt{21} \right)}\right) - \pi \right)} < \sqrt{21}

    /1        /  ____\\     ____
-cot|-- - acot\\/ 21 /| < \/ 21 
    \10               /

but

    /1        /  ____\\     ____
-cot|-- - acot\\/ 21 /| > \/ 21 
    \10               /

Then

x < \pi n + \operatorname{acot}{\left(\sqrt{21} \right)}

no execute
the solution of our inequality is:

x > \pi n + \operatorname{acot}{\left(\sqrt{21} \right)}

         _____  
        /
-------ο-------
       x_1

Solving inequality on a graph

The graph