Mister Exam
tg(x/2)⩽1 inequation
The solution
Detail solution
Given the inequality:
$$\tan{\left(\frac{x}{2} \right)} \leq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(\frac{x}{2} \right)} = 1$$
Solve:
Given the equation
$$\tan{\left(\frac{x}{2} \right)} = 1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{2} = \pi n + \operatorname{atan}{\left(1 \right)}$$
Or
$$\frac{x}{2} = \pi n + \frac{\pi}{4}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
This roots
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n + \frac{\pi}{2}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10} + \frac{\pi}{2}$$
substitute to the expression
$$\tan{\left(\frac{x}{2} \right)} \leq 1$$
$$\tan{\left(\frac{2 \pi n - \frac{1}{10} + \frac{\pi}{2}}{2} \right)} \leq 1$$
the solution of our inequality is:
$$x \leq 2 \pi n + \frac{\pi}{2}$$
$$\tan{\left(\frac{x}{2} \right)} \leq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(\frac{x}{2} \right)} = 1$$
Solve:
Given the equation
$$\tan{\left(\frac{x}{2} \right)} = 1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{2} = \pi n + \operatorname{atan}{\left(1 \right)}$$
Or
$$\frac{x}{2} = \pi n + \frac{\pi}{4}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
This roots
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n + \frac{\pi}{2}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10} + \frac{\pi}{2}$$
substitute to the expression
$$\tan{\left(\frac{x}{2} \right)} \leq 1$$
$$\tan{\left(\frac{2 \pi n - \frac{1}{10} + \frac{\pi}{2}}{2} \right)} \leq 1$$
/ 1 pi \ tan|- -- + -- + pi*n| <= 1 \ 20 4 /
the solution of our inequality is:
$$x \leq 2 \pi n + \frac{\pi}{2}$$
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x1
Solving inequality on a graph
Rapid solution 2
[src]
pi
[0, --] U (pi, 2*pi]
2
$$x\ in\ \left[0, \frac{\pi}{2}\right] \cup \left(\pi, 2 \pi\right]$$
x in Union(Interval(0, pi/2), Interval.Lopen(pi, 2*pi))
Rapid solution
[src]
/ / pi\ \ Or|And|0 <= x, x <= --|, And(x <= 2*pi, pi < x)| \ \ 2 / /
$$\left(0 \leq x \wedge x \leq \frac{\pi}{2}\right) \vee \left(x \leq 2 \pi \wedge \pi < x\right)$$
((0 <= x)∧(x <= pi/2))∨((pi < x)∧(x <= 2*pi))