Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- |0,5x+1|<=0
- |x-1|-(6/|x-1|)<1
-
64/(2^(x+3))<=0,125^x-7
- tg^3x+3>tgx+tg^2x
- Derivative of:
-
tg(2x)
- Integral of d{x}:
-
tg(2x)
-
Identical expressions
- tg(2x)<= one
- tg(2x) less than or equal to 1
- tg(2x) less than or equal to one
- tg2x<=1
-
Similar expressions
- ctg^2(x)<ctg(x)
- arctg(2x^2-5x-1)+arctg(x^2-2x+5)≤0
- tg(2*x-(pi/3))<sqrt(3)/3
- tg^3x+3>tgx+tg^2x
tg(2x)<=1 inequation
The solution
Detail solution
Given the inequality:
$$\tan{\left(2 x \right)} \leq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(2 x \right)} = 1$$
Solve:
Given the equation
$$\tan{\left(2 x \right)} = 1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$2 x = \pi n + \operatorname{atan}{\left(1 \right)}$$
Or
$$2 x = \pi n + \frac{\pi}{4}$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
$$x_{1} = \frac{\pi n}{2} + \frac{\pi}{8}$$
$$x_{1} = \frac{\pi n}{2} + \frac{\pi}{8}$$
This roots
$$x_{1} = \frac{\pi n}{2} + \frac{\pi}{8}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{\pi n}{2} + \frac{\pi}{8}\right) + - \frac{1}{10}$$
=
$$\frac{\pi n}{2} - \frac{1}{10} + \frac{\pi}{8}$$
substitute to the expression
$$\tan{\left(2 x \right)} \leq 1$$
$$\tan{\left(2 \left(\frac{\pi n}{2} - \frac{1}{10} + \frac{\pi}{8}\right) \right)} \leq 1$$
the solution of our inequality is:
$$x \leq \frac{\pi n}{2} + \frac{\pi}{8}$$
$$\tan{\left(2 x \right)} \leq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(2 x \right)} = 1$$
Solve:
Given the equation
$$\tan{\left(2 x \right)} = 1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$2 x = \pi n + \operatorname{atan}{\left(1 \right)}$$
Or
$$2 x = \pi n + \frac{\pi}{4}$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
$$x_{1} = \frac{\pi n}{2} + \frac{\pi}{8}$$
$$x_{1} = \frac{\pi n}{2} + \frac{\pi}{8}$$
This roots
$$x_{1} = \frac{\pi n}{2} + \frac{\pi}{8}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{\pi n}{2} + \frac{\pi}{8}\right) + - \frac{1}{10}$$
=
$$\frac{\pi n}{2} - \frac{1}{10} + \frac{\pi}{8}$$
substitute to the expression
$$\tan{\left(2 x \right)} \leq 1$$
$$\tan{\left(2 \left(\frac{\pi n}{2} - \frac{1}{10} + \frac{\pi}{8}\right) \right)} \leq 1$$
/ 1 pi \ tan|- - + -- + pi*n| <= 1 \ 5 4 /
the solution of our inequality is:
$$x \leq \frac{\pi n}{2} + \frac{\pi}{8}$$
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x1
Solving inequality on a graph
Rapid solution 2
[src]
/ ___________\
| / ___ |
|\/ 2 - \/ 2 | pi pi
[0, atan|--------------|] U (--, --]
| ___________| 4 2
| / ___ |
\\/ 2 + \/ 2 /
$$x\ in\ \left[0, \operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)}\right] \cup \left(\frac{\pi}{4}, \frac{\pi}{2}\right]$$
x in Union(Interval(0, atan(sqrt(2 - sqrt(2))/sqrt(sqrt(2) + 2))), Interval.Lopen(pi/4, pi/2))
Rapid solution
[src]
/ / / ___________\\ \ | | | / ___ || | | | |\/ 2 - \/ 2 || / pi pi \| Or|And|0 <= x, x <= atan|--------------||, And|x <= --, -- < x|| | | | ___________|| \ 2 4 /| | | | / ___ || | \ \ \\/ 2 + \/ 2 // /
$$\left(0 \leq x \wedge x \leq \operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)}\right) \vee \left(x \leq \frac{\pi}{2} \wedge \frac{\pi}{4} < x\right)$$
((x <= pi/2)∧(pi/4 < x))∨((0 <= x)∧(x <= atan(sqrt(2 - sqrt(2))/sqrt(2 + sqrt(2)))))