Mister Exam
tan((x+pi)/3)>0 inequation
The solution
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/x + pi\ tan|------| > 0 \ 3 /
$$\tan{\left(\frac{x + \pi}{3} \right)} > 0$$
tan((x + pi)/3) > 0
Detail solution
Given the inequality:
$$\tan{\left(\frac{x + \pi}{3} \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(\frac{x + \pi}{3} \right)} = 0$$
Solve:
Given the equation
$$\tan{\left(\frac{x + \pi}{3} \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\tan{\left(\frac{x + \pi}{3} \right)} = 0$$
This equation is transformed to
$$\frac{x}{3} + \frac{\pi}{3} = \pi n + \operatorname{atan}{\left(0 \right)}$$
Or
$$\frac{x}{3} + \frac{\pi}{3} = \pi n$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{3} = \pi n - \frac{\pi}{3}$$
Divide both parts of the equation by
$$\frac{1}{3}$$
$$x_{1} = 3 \pi n - \pi$$
$$x_{1} = 3 \pi n - \pi$$
This roots
$$x_{1} = 3 \pi n - \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(3 \pi n - \pi\right) + - \frac{1}{10}$$
=
$$3 \pi n - \pi - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(\frac{x + \pi}{3} \right)} > 0$$
$$\tan{\left(\frac{\left(3 \pi n - \pi - \frac{1}{10}\right) + \pi}{3} \right)} > 0$$
Then
$$x < 3 \pi n - \pi$$
no execute
the solution of our inequality is:
$$x > 3 \pi n - \pi$$
$$\tan{\left(\frac{x + \pi}{3} \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(\frac{x + \pi}{3} \right)} = 0$$
Solve:
Given the equation
$$\tan{\left(\frac{x + \pi}{3} \right)} = 0$$
- this is the simplest trigonometric equation
with the change of sign in 0
We get:
$$\tan{\left(\frac{x + \pi}{3} \right)} = 0$$
This equation is transformed to
$$\frac{x}{3} + \frac{\pi}{3} = \pi n + \operatorname{atan}{\left(0 \right)}$$
Or
$$\frac{x}{3} + \frac{\pi}{3} = \pi n$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{3} = \pi n - \frac{\pi}{3}$$
Divide both parts of the equation by
$$\frac{1}{3}$$
$$x_{1} = 3 \pi n - \pi$$
$$x_{1} = 3 \pi n - \pi$$
This roots
$$x_{1} = 3 \pi n - \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(3 \pi n - \pi\right) + - \frac{1}{10}$$
=
$$3 \pi n - \pi - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(\frac{x + \pi}{3} \right)} > 0$$
$$\tan{\left(\frac{\left(3 \pi n - \pi - \frac{1}{10}\right) + \pi}{3} \right)} > 0$$
tan(-1/30 + pi*n) > 0
Then
$$x < 3 \pi n - \pi$$
no execute
the solution of our inequality is:
$$x > 3 \pi n - \pi$$
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Solving inequality on a graph
Rapid solution 2
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pi
[0, --) U (2*pi, 3*pi]
2
$$x\ in\ \left[0, \frac{\pi}{2}\right) \cup \left(2 \pi, 3 \pi\right]$$
x in Union(Interval.Ropen(0, pi/2), Interval.Lopen(2*pi, 3*pi))
Rapid solution
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/ / pi\ \ Or|And|0 <= x, x < --|, And(x <= 3*pi, 2*pi < x)| \ \ 2 / /
$$\left(0 \leq x \wedge x < \frac{\pi}{2}\right) \vee \left(x \leq 3 \pi \wedge 2 \pi < x\right)$$
((0 <= x)∧(x < pi/2))∨((x <= 3*pi)∧(2*pi < x))