Mister Exam
cot(x-2*pi/3)>=(-sqrt(3))/3 inequation
The solution
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___ / 2*pi\ -\/ 3 cot|x - ----| >= ------- \ 3 / 3
$$\cot{\left(x - \frac{2 \pi}{3} \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
cot(x - 2*pi/3) >= -sqrt(3)/3
Detail solution
Given the inequality:
$$\cot{\left(x - \frac{2 \pi}{3} \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cot{\left(x - \frac{2 \pi}{3} \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
Solve:
Given the equation
$$\cot{\left(x - \frac{2 \pi}{3} \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x + \frac{\pi}{3} = \pi n + \operatorname{acot}{\left(- \frac{\sqrt{3}}{3} \right)}$$
Or
$$x + \frac{\pi}{3} = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation with the opposite sign, in total:
$$x = \pi n - \frac{2 \pi}{3}$$
$$x_{1} = \pi n - \frac{2 \pi}{3}$$
$$x_{1} = \pi n - \frac{2 \pi}{3}$$
This roots
$$x_{1} = \pi n - \frac{2 \pi}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n - \frac{2 \pi}{3}\right) - \frac{1}{10}$$
=
$$\pi n - \frac{2 \pi}{3} - \frac{1}{10}$$
substitute to the expression
$$\cot{\left(x - \frac{2 \pi}{3} \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
$$\cot{\left(\left(\pi n - \frac{2 \pi}{3} - \frac{1}{10}\right) - \frac{2 \pi}{3} \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
the solution of our inequality is:
$$x \leq \pi n - \frac{2 \pi}{3}$$
$$\cot{\left(x - \frac{2 \pi}{3} \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cot{\left(x - \frac{2 \pi}{3} \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
Solve:
Given the equation
$$\cot{\left(x - \frac{2 \pi}{3} \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x + \frac{\pi}{3} = \pi n + \operatorname{acot}{\left(- \frac{\sqrt{3}}{3} \right)}$$
Or
$$x + \frac{\pi}{3} = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation with the opposite sign, in total:
$$x = \pi n - \frac{2 \pi}{3}$$
$$x_{1} = \pi n - \frac{2 \pi}{3}$$
$$x_{1} = \pi n - \frac{2 \pi}{3}$$
This roots
$$x_{1} = \pi n - \frac{2 \pi}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n - \frac{2 \pi}{3}\right) - \frac{1}{10}$$
=
$$\pi n - \frac{2 \pi}{3} - \frac{1}{10}$$
substitute to the expression
$$\cot{\left(x - \frac{2 \pi}{3} \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
$$\cot{\left(\left(\pi n - \frac{2 \pi}{3} - \frac{1}{10}\right) - \frac{2 \pi}{3} \right)} \geq \frac{\left(-1\right) \sqrt{3}}{3}$$
___
/1 pi\ -\/ 3
-cot|-- + --| >= -------
\10 3 / 3
the solution of our inequality is:
$$x \leq \pi n - \frac{2 \pi}{3}$$
_____
\
-------•-------
x_1
Rapid solution
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/ / pi\ /2*pi \\ Or|And|0 <= x, x <= --|, And|---- < x, x < pi|| \ \ 3 / \ 3 //
$$\left(0 \leq x \wedge x \leq \frac{\pi}{3}\right) \vee \left(\frac{2 \pi}{3} < x \wedge x < \pi\right)$$
((0 <= x)∧(x <= pi/3))∨((x < pi)∧(2*pi/3 < x))
Rapid solution 2
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pi 2*pi
[0, --] U (----, pi)
3 3
$$x\ in\ \left[0, \frac{\pi}{3}\right] \cup \left(\frac{2 \pi}{3}, \pi\right)$$
x in Union(Interval(0, pi/3), Interval.open(2*pi/3, pi))