tg(3x)

+ inequation

Solve the inequality!

A inequation with variable

The solution

You have entered [src]

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tan(3*x) < \/ 3 

$$\tan{\left(3 x \right)} < \sqrt{3}$$

tan(3*x) < sqrt(3)

Detail solution

Given the inequality:
$$\tan{\left(3 x \right)} < \sqrt{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(3 x \right)} = \sqrt{3}$$
Solve:
Given the equation
$$\tan{\left(3 x \right)} = \sqrt{3}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$3 x = \pi n + \operatorname{atan}{\left(\sqrt{3} \right)}$$
Or
$$3 x = \pi n + \frac{\pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$3$$
$$x_{1} = \frac{\pi n}{3} + \frac{\pi}{9}$$
$$x_{1} = \frac{\pi n}{3} + \frac{\pi}{9}$$
This roots
$$x_{1} = \frac{\pi n}{3} + \frac{\pi}{9}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{\pi n}{3} + \frac{\pi}{9}\right) - \frac{1}{10}$$
=
$$\frac{\pi n}{3} - \frac{1}{10} + \frac{\pi}{9}$$
substitute to the expression
$$\tan{\left(3 x \right)} < \sqrt{3}$$
$$\tan{\left(3 \left(\frac{\pi n}{3} - \frac{1}{10} + \frac{\pi}{9}\right) \right)} < \sqrt{3}$$

   /3    pi\     ___
cot|-- + --| < \/ 3 
   \10   6 /   

the solution of our inequality is:
$$x < \frac{\pi n}{3} + \frac{\pi}{9}$$

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       x_1

Solving inequality on a graph

Rapid solution [src]

  /   /            pi\     /pi          pi\\
Or|And|0 <= x, x < --|, And|-- < x, x < --||
  \   \            9 /     \6           3 //

$$\left(0 \leq x \wedge x < \frac{\pi}{9}\right) \vee \left(\frac{\pi}{6} < x \wedge x < \frac{\pi}{3}\right)$$

((0 <= x)∧(x < pi/9))∨((pi/6 < x)∧(x < pi/3))

Rapid solution 2 [src]

    pi     pi  pi 
[0, --) U (--, --)
    9      6   3  

$$x\ in\ \left[0, \frac{\pi}{9}\right) \cup \left(\frac{\pi}{6}, \frac{\pi}{3}\right)$$

x in Union(Interval.Ropen(0, pi/9), Interval.open(pi/6, pi/3))

The graph