Mister Exam
tgx≤√3/3 inequation
The solution
You have entered
[src]
___
\/ 3
tan(x) <= -----
3
$$\tan{\left(x \right)} \leq \frac{\sqrt{3}}{3}$$
tan(x) <= sqrt(3)/3
Detail solution
Given the inequality:
$$\tan{\left(x \right)} \leq \frac{\sqrt{3}}{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(x \right)} = \frac{\sqrt{3}}{3}$$
Solve:
Given the equation
$$\tan{\left(x \right)} = \frac{\sqrt{3}}{3}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(\frac{\sqrt{3}}{3} \right)}$$
Or
$$x = \pi n + \frac{\pi}{6}$$
, 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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \frac{\pi}{6}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \frac{\pi}{6}$$
substitute to the expression
$$\tan{\left(x \right)} \leq \frac{\sqrt{3}}{3}$$
$$\tan{\left(\pi n - \frac{1}{10} + \frac{\pi}{6} \right)} \leq \frac{\sqrt{3}}{3}$$
the solution of our inequality is:
$$x \leq \pi n + \frac{\pi}{6}$$
$$\tan{\left(x \right)} \leq \frac{\sqrt{3}}{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(x \right)} = \frac{\sqrt{3}}{3}$$
Solve:
Given the equation
$$\tan{\left(x \right)} = \frac{\sqrt{3}}{3}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(\frac{\sqrt{3}}{3} \right)}$$
Or
$$x = \pi n + \frac{\pi}{6}$$
, 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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \frac{\pi}{6}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \frac{\pi}{6}$$
substitute to the expression
$$\tan{\left(x \right)} \leq \frac{\sqrt{3}}{3}$$
$$\tan{\left(\pi n - \frac{1}{10} + \frac{\pi}{6} \right)} \leq \frac{\sqrt{3}}{3}$$
___
/ 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]
/ / pi\ / pi \\ Or|And|0 <= x, x <= --|, And|x <= pi, -- < x|| \ \ 6 / \ 2 //
$$\left(0 \leq x \wedge x \leq \frac{\pi}{6}\right) \vee \left(x \leq \pi \wedge \frac{\pi}{2} < x\right)$$
((0 <= x)∧(x <= pi/6))∨((x <= pi)∧(pi/2 < x))
Rapid solution 2
[src]
pi pi
[0, --] U (--, pi]
6 2
$$x\ in\ \left[0, \frac{\pi}{6}\right] \cup \left(\frac{\pi}{2}, \pi\right]$$
x in Union(Interval(0, pi/6), Interval.Lopen(pi/2, pi))