Mister Exam
(2sin(x)-3)tgx>0 inequation
The solution
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(2*sin(x) - 3)*tan(x) > 0
$$\left(2 \sin{\left(x \right)} - 3\right) \tan{\left(x \right)} > 0$$
(2*sin(x) - 3)*tan(x) > 0
Detail solution
Given the inequality:
$$\left(2 \sin{\left(x \right)} - 3\right) \tan{\left(x \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(2 \sin{\left(x \right)} - 3\right) \tan{\left(x \right)} = 0$$
Solve:
$$x_{1} = 0$$
$$x_{2} = 2 \operatorname{atan}{\left(\frac{2}{3} - \frac{\sqrt{5} i}{3} \right)}$$
$$x_{3} = 2 \operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}$$
Exclude the complex solutions:
$$x_{1} = 0$$
This roots
$$x_{1} = 0$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10}$$
=
$$- \frac{1}{10}$$
substitute to the expression
$$\left(2 \sin{\left(x \right)} - 3\right) \tan{\left(x \right)} > 0$$
$$\left(-3 + 2 \sin{\left(- \frac{1}{10} \right)}\right) \tan{\left(- \frac{1}{10} \right)} > 0$$
the solution of our inequality is:
$$x < 0$$
$$\left(2 \sin{\left(x \right)} - 3\right) \tan{\left(x \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(2 \sin{\left(x \right)} - 3\right) \tan{\left(x \right)} = 0$$
Solve:
$$x_{1} = 0$$
$$x_{2} = 2 \operatorname{atan}{\left(\frac{2}{3} - \frac{\sqrt{5} i}{3} \right)}$$
$$x_{3} = 2 \operatorname{atan}{\left(\frac{2}{3} + \frac{\sqrt{5} i}{3} \right)}$$
Exclude the complex solutions:
$$x_{1} = 0$$
This roots
$$x_{1} = 0$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10}$$
=
$$- \frac{1}{10}$$
substitute to the expression
$$\left(2 \sin{\left(x \right)} - 3\right) \tan{\left(x \right)} > 0$$
$$\left(-3 + 2 \sin{\left(- \frac{1}{10} \right)}\right) \tan{\left(- \frac{1}{10} \right)} > 0$$
-(-3 - 2*sin(1/10))*tan(1/10) > 0
the solution of our inequality is:
$$x < 0$$
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Solving inequality on a graph
Rapid solution
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/ /3*pi \ /pi \\ Or|And|---- < x, x < 2*pi|, And|-- < x, x < pi|| \ \ 2 / \2 //
$$\left(\frac{3 \pi}{2} < x \wedge x < 2 \pi\right) \vee \left(\frac{\pi}{2} < x \wedge x < \pi\right)$$
((x < pi)∧(pi/2 < x))∨((3*pi/2 < x)∧(x < 2*pi))
Rapid solution 2
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pi 3*pi (--, pi) U (----, 2*pi) 2 2
$$x\ in\ \left(\frac{\pi}{2}, \pi\right) \cup \left(\frac{3 \pi}{2}, 2 \pi\right)$$
x in Union(Interval.open(pi/2, pi), Interval.open(3*pi/2, 2*pi))