Mister Exam
cbrtctg(4x-pi/9)》0 inequation
The solution
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_______________ / / pi\ 3 / cot|4*x - --| > 0 \/ \ 9 /
$$\sqrt[3]{\cot{\left(4 x - \frac{\pi}{9} \right)}} > 0$$
cot(4*x - pi/9)^(1/3) > 0
Detail solution
Given the inequality:
$$\sqrt[3]{\cot{\left(4 x - \frac{\pi}{9} \right)}} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\sqrt[3]{\cot{\left(4 x - \frac{\pi}{9} \right)}} = 0$$
Solve:
Given the equation
$$\sqrt[3]{\cot{\left(4 x - \frac{\pi}{9} \right)}} = 0$$
transform
$$\sqrt[3]{- \tan{\left(4 x + \frac{7 \pi}{18} \right)}} = 0$$
$$\sqrt[3]{\cot{\left(4 x - \frac{\pi}{9} \right)}} = 0$$
Do replacement
$$w = \tan{\left(4 x + \frac{7 \pi}{18} \right)}$$
Given the equation
$$\sqrt[3]{\cot{\left(4 x - \frac{\pi}{9} \right)}} = 0$$
so
$$- \tan{\left(4 x + \frac{7 \pi}{18} \right)} = 0$$
Expand brackets in the left part
This equation has no roots
do backward replacement
$$\tan{\left(4 x + \frac{7 \pi}{18} \right)} = w$$
Given the equation
$$\tan{\left(4 x + \frac{7 \pi}{18} \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$4 x + \frac{7 \pi}{18} = \pi n + \operatorname{atan}{\left(w \right)}$$
Or
$$4 x + \frac{7 \pi}{18} = \pi n + \operatorname{atan}{\left(w \right)}$$
, where n - is a integer
Move
$$\frac{7 \pi}{18}$$
to right part of the equation
with the opposite sign, in total:
$$4 x = \pi n + \operatorname{atan}{\left(w \right)} - \frac{7 \pi}{18}$$
Divide both parts of the equation by
$$4$$
substitute w:
$$x_{1} = - \frac{7 \pi}{72}$$
$$x_{1} = - \frac{7 \pi}{72}$$
This roots
$$x_{1} = - \frac{7 \pi}{72}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{7 \pi}{72} - \frac{1}{10}$$
=
$$- \frac{7 \pi}{72} - \frac{1}{10}$$
substitute to the expression
$$\sqrt[3]{\cot{\left(4 x - \frac{\pi}{9} \right)}} > 0$$
$$\sqrt[3]{\cot{\left(4 \left(- \frac{7 \pi}{72} - \frac{1}{10}\right) - \frac{\pi}{9} \right)}} > 0$$
the solution of our inequality is:
$$x < - \frac{7 \pi}{72}$$
$$\sqrt[3]{\cot{\left(4 x - \frac{\pi}{9} \right)}} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\sqrt[3]{\cot{\left(4 x - \frac{\pi}{9} \right)}} = 0$$
Solve:
Given the equation
$$\sqrt[3]{\cot{\left(4 x - \frac{\pi}{9} \right)}} = 0$$
transform
$$\sqrt[3]{- \tan{\left(4 x + \frac{7 \pi}{18} \right)}} = 0$$
$$\sqrt[3]{\cot{\left(4 x - \frac{\pi}{9} \right)}} = 0$$
Do replacement
$$w = \tan{\left(4 x + \frac{7 \pi}{18} \right)}$$
Given the equation
$$\sqrt[3]{\cot{\left(4 x - \frac{\pi}{9} \right)}} = 0$$
so
$$- \tan{\left(4 x + \frac{7 \pi}{18} \right)} = 0$$
Expand brackets in the left part
-tan4*x+7*pi/18 = 0
This equation has no roots
do backward replacement
$$\tan{\left(4 x + \frac{7 \pi}{18} \right)} = w$$
Given the equation
$$\tan{\left(4 x + \frac{7 \pi}{18} \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$4 x + \frac{7 \pi}{18} = \pi n + \operatorname{atan}{\left(w \right)}$$
Or
$$4 x + \frac{7 \pi}{18} = \pi n + \operatorname{atan}{\left(w \right)}$$
, where n - is a integer
Move
$$\frac{7 \pi}{18}$$
to right part of the equation
with the opposite sign, in total:
$$4 x = \pi n + \operatorname{atan}{\left(w \right)} - \frac{7 \pi}{18}$$
Divide both parts of the equation by
$$4$$
substitute w:
$$x_{1} = - \frac{7 \pi}{72}$$
$$x_{1} = - \frac{7 \pi}{72}$$
This roots
$$x_{1} = - \frac{7 \pi}{72}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{7 \pi}{72} - \frac{1}{10}$$
=
$$- \frac{7 \pi}{72} - \frac{1}{10}$$
substitute to the expression
$$\sqrt[3]{\cot{\left(4 x - \frac{\pi}{9} \right)}} > 0$$
$$\sqrt[3]{\cot{\left(4 \left(- \frac{7 \pi}{72} - \frac{1}{10}\right) - \frac{\pi}{9} \right)}} > 0$$
3 __________
\/ tan(2/5) > 0
the solution of our inequality is:
$$x < - \frac{7 \pi}{72}$$
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