Mister Exam

Lang:

tg(x)<=-√3/3 inequation

You have entered [src]

             ___ 
          -\/ 3  
tan(x) <= -------
             3

\tan{\left(x \right)} \leq \frac{\left(-1\right) \sqrt{3}}{3}

tan(x) <= (-sqrt(3))/3

Detail solution

Given the inequality:

\tan{\left(x \right)} \leq \frac{\left(-1\right) \sqrt{3}}{3}

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

\tan{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{3}

Solve:
Given the equation

\tan{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{3}

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

x = \pi n + \operatorname{atan}{\left(- \frac{\sqrt{3}}{3} \right)}

x = \pi n - \frac{\pi}{6}

, where n - is a integer

x_{1} = \pi n - \frac{\pi}{6}

x_{1} = \pi n - \frac{\pi}{6}

This roots

x_{1} = \pi n - \frac{\pi}{6}

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}{6}\right) + - \frac{1}{10}

\pi n - \frac{\pi}{6} - \frac{1}{10}

substitute to the expression

\tan{\left(x \right)} \leq \frac{\left(-1\right) \sqrt{3}}{3}

\tan{\left(\pi n - \frac{\pi}{6} - \frac{1}{10} \right)} \leq \frac{\left(-1\right) \sqrt{3}}{3}

                           ___ 
    /1    pi       \    -\/ 3  
-tan|-- + -- - pi*n| <= -------
    \10   6        /       3

the solution of our inequality is:

x \leq \pi n - \frac{\pi}{6}

 _____          
      \    
-------•-------
       x1

Solving inequality on a graph

Rapid solution [src]

   /     5*pi  pi    \
And|x <= ----, -- < x|
   \      6    2     /

x \leq \frac{5 \pi}{6} \wedge \frac{\pi}{2} < x

(x <= 5*pi/6)∧(pi/2 < x)

Rapid solution 2 [src]

 pi  5*pi 
(--, ----]
 2    6

x\ in\ \left(\frac{\pi}{2}, \frac{5 \pi}{6}\right]

x in Interval.Lopen(pi/2, 5*pi/6)