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Solving inequalities/ tgx>=3/3

tgx>=3/3 inequation

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tan(x) >= 1

\tan{\left(x \right)} \geq 1

tan(x) >= 1

Detail solution

Given the inequality:

\tan{\left(x \right)} \geq 1

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

\tan{\left(x \right)} = 1

Solve:
Given the equation

\tan{\left(x \right)} = 1

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

x = \pi n + \operatorname{atan}{\left(1 \right)}

x = \pi n + \frac{\pi}{4}

, where n - is a integer

x_{1} = \pi n + \frac{\pi}{4}

x_{1} = \pi n + \frac{\pi}{4}

This roots

x_{1} = \pi n + \frac{\pi}{4}

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

x_{0} \leq x_{1}

For example, let's take the point

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

\left(\pi n + \frac{\pi}{4}\right) - \frac{1}{10}

\pi n - \frac{1}{10} + \frac{\pi}{4}

substitute to the expression

\tan{\left(x \right)} \geq 1

\tan{\left(\pi n - \frac{1}{10} + \frac{\pi}{4} \right)} \geq 1

   /1    pi\     
cot|-- + --| >= 1
   \10   4 /

but

   /1    pi\    
cot|-- + --| < 1
   \10   4 /

Then

x \leq \pi n + \frac{\pi}{4}

no execute
the solution of our inequality is:

x \geq \pi n + \frac{\pi}{4}

         _____  
        /
-------•-------
       x_1

Solving inequality on a graph

Rapid solution [src]

   /pi           pi\
And|-- <= x, x < --|
   \4            2 /

\frac{\pi}{4} \leq x \wedge x < \frac{\pi}{2}

(pi/4 <= x)∧(x < pi/2)

Rapid solution 2 [src]

 pi  pi 
[--, --)
 4   2

x\ in\ \left[\frac{\pi}{4}, \frac{\pi}{2}\right)

x in Interval.Ropen(pi/4, pi/2)

The graph