Mister Exam
tg(x)≥-√3/3 inequation
The solution
You have entered
[src]
___
-\/ 3
tan(x) >= -------
3
$$\tan{\left(x \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
tan(x) >= (-sqrt(3))/3
Detail solution
Given the inequality:
$$\tan{\left(x \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
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
$$x = \pi n + \operatorname{atan}{\left(- \frac{\sqrt{3}}{3} \right)}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n - \frac{\pi}{6}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{\pi}{6} - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(x \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
$$\tan{\left(\pi n - \frac{\pi}{6} - \frac{1}{10} \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
but
Then
$$x \leq \pi n - \frac{\pi}{6}$$
no execute
the solution of our inequality is:
$$x \geq \pi n - \frac{\pi}{6}$$
$$\tan{\left(x \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
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
$$x = \pi n + \operatorname{atan}{\left(- \frac{\sqrt{3}}{3} \right)}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n - \frac{\pi}{6}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{\pi}{6} - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(x \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
$$\tan{\left(\pi n - \frac{\pi}{6} - \frac{1}{10} \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
___
/1 pi \ -\/ 3
-tan|-- + -- - pi*n| >= -------
\10 6 / 3
but
___
/1 pi \ -\/ 3
-tan|-- + -- - pi*n| < -------
\10 6 / 3
Then
$$x \leq \pi n - \frac{\pi}{6}$$
no execute
the solution of our inequality is:
$$x \geq \pi n - \frac{\pi}{6}$$
_____
/
-------•-------
x1
Solving inequality on a graph
Rapid solution
[src]
/ / pi\ /5*pi \\ Or|And|0 <= x, x < --|, And|---- <= x, x <= pi|| \ \ 2 / \ 6 //
$$\left(0 \leq x \wedge x < \frac{\pi}{2}\right) \vee \left(\frac{5 \pi}{6} \leq x \wedge x \leq \pi\right)$$
((0 <= x)∧(x < pi/2))∨((x <= pi)∧(5*pi/6 <= x))
Rapid solution 2
[src]
pi 5*pi
[0, --) U [----, pi]
2 6
$$x\ in\ \left[0, \frac{\pi}{2}\right) \cup \left[\frac{5 \pi}{6}, \pi\right]$$
x in Union(Interval.Ropen(0, pi/2), Interval(5*pi/6, pi))