Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- 18>=x
- 2x^2-4x+1>=0
- tan(2x)<=1/2
- 7^(x²+5x)>1
- Derivative of:
-
tan(2x)
- Integral of d{x}:
-
tan(2x)
-
Identical expressions
- tan(two x)<= one /2
- tangent of (2x) less than or equal to 1 divide by 2
- tangent of (two x) less than or equal to one divide by 2
- tan2x<=1/2
- tan(2x)<=1 divide by 2
-
Similar expressions
- atan(2^x)<1
- tan^2x≤0
tan(2x)<=1/2 inequation
The solution
Detail solution
Given the inequality:
$$\tan{\left(2 x \right)} \leq \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(2 x \right)} = \frac{1}{2}$$
Solve:
Given the equation
$$\tan{\left(2 x \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$2 x = \pi n + \operatorname{atan}{\left(\frac{1}{2} \right)}$$
Or
$$2 x = \pi n + \operatorname{atan}{\left(\frac{1}{2} \right)}$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2}$$
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2}$$
This roots
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{\pi n}{2} + \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2}\right) + - \frac{1}{10}$$
=
$$\frac{\pi n}{2} - \frac{1}{10} + \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2}$$
substitute to the expression
$$\tan{\left(2 x \right)} \leq \frac{1}{2}$$
$$\tan{\left(2 \left(\frac{\pi n}{2} - \frac{1}{10} + \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2}\right) \right)} \leq \frac{1}{2}$$
the solution of our inequality is:
$$x \leq \frac{\pi n}{2} + \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2}$$
$$\tan{\left(2 x \right)} \leq \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(2 x \right)} = \frac{1}{2}$$
Solve:
Given the equation
$$\tan{\left(2 x \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$2 x = \pi n + \operatorname{atan}{\left(\frac{1}{2} \right)}$$
Or
$$2 x = \pi n + \operatorname{atan}{\left(\frac{1}{2} \right)}$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2}$$
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2}$$
This roots
$$x_{1} = \frac{\pi n}{2} + \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{\pi n}{2} + \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2}\right) + - \frac{1}{10}$$
=
$$\frac{\pi n}{2} - \frac{1}{10} + \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2}$$
substitute to the expression
$$\tan{\left(2 x \right)} \leq \frac{1}{2}$$
$$\tan{\left(2 \left(\frac{\pi n}{2} - \frac{1}{10} + \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2}\right) \right)} \leq \frac{1}{2}$$
tan(-1/5 + pi*n + atan(1/2)) <= 1/2
the solution of our inequality is:
$$x \leq \frac{\pi n}{2} + \frac{\operatorname{atan}{\left(\frac{1}{2} \right)}}{2}$$
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Solving inequality on a graph
Rapid solution
[src]
/ / / / /atan(4/3)\\ \\ \ | | | |sin|---------|| / ___________________________________\|| | | | | | \ 4 /| | / 2/atan(4/3)\ 2/atan(4/3)\ ||| / pi pi \| Or|And|0 <= x, x <= -I*|I*atan|--------------| + log| / cos |---------| + sin |---------| |||, And|x <= --, -- < x|| | | | | /atan(4/3)\| \\/ \ 4 / \ 4 / /|| \ 2 4 /| | | | |cos|---------|| || | \ \ \ \ \ 4 // // /
$$\left(0 \leq x \wedge x \leq - i \left(\log{\left(\sqrt{\sin^{2}{\left(\frac{\operatorname{atan}{\left(\frac{4}{3} \right)}}{4} \right)} + \cos^{2}{\left(\frac{\operatorname{atan}{\left(\frac{4}{3} \right)}}{4} \right)}} \right)} + i \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{4}{3} \right)}}{4} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(\frac{4}{3} \right)}}{4} \right)}} \right)}\right)\right) \vee \left(x \leq \frac{\pi}{2} \wedge \frac{\pi}{4} < x\right)$$
((x <= pi/2)∧(pi/4 < x))∨((0 <= x)∧(x <= -i*(i*atan(sin(atan(4/3)/4)/cos(atan(4/3)/4)) + log(sqrt(cos(atan(4/3)/4)^2 + sin(atan(4/3)/4)^2)))))