Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- 3^x>7*2^x
- (x+10)*1/2>x-2
- -7x^2+5x-2<0
- (x+1)/x-(x-1)/(x+1)<2
- Graphing y =:
-
sin2x
- Derivative of:
-
sin2x
- Sum of series:
- sin2x
-
Identical expressions
- sin two x<=√ two \2
- sinus of 2x less than or equal to √2\2
- sinus of two x less than or equal to √ two \2
-
Similar expressions
- 2sin(2x)<=1
- 2sin^2x−sinx−3<0
sin2x<=√2\2 inequation
The solution
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___
\/ 2
sin(2*x) <= -----
2
$$\sin{\left(2 x \right)} \leq \frac{\sqrt{2}}{2}$$
sin(2*x) <= sqrt(2)/2
Detail solution
Given the inequality:
$$\sin{\left(2 x \right)} \leq \frac{\sqrt{2}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(2 x \right)} = \frac{\sqrt{2}}{2}$$
Solve:
Given the equation
$$\sin{\left(2 x \right)} = \frac{\sqrt{2}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$2 x = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$2 x = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
Or
$$2 x = 2 \pi n + \frac{\pi}{4}$$
$$2 x = 2 \pi n + \frac{3 \pi}{4}$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
$$x_{1} = \pi n + \frac{\pi}{8}$$
$$x_{2} = \pi n + \frac{3 \pi}{8}$$
$$x_{1} = \pi n + \frac{\pi}{8}$$
$$x_{2} = \pi n + \frac{3 \pi}{8}$$
This roots
$$x_{1} = \pi n + \frac{\pi}{8}$$
$$x_{2} = \pi n + \frac{3 \pi}{8}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \frac{\pi}{8}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \frac{\pi}{8}$$
substitute to the expression
$$\sin{\left(2 x \right)} \leq \frac{\sqrt{2}}{2}$$
$$\sin{\left(2 \left(\pi n - \frac{1}{10} + \frac{\pi}{8}\right) \right)} \leq \frac{\sqrt{2}}{2}$$
one of the solutions of our inequality is:
$$x \leq \pi n + \frac{\pi}{8}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq \pi n + \frac{\pi}{8}$$
$$x \geq \pi n + \frac{3 \pi}{8}$$
$$\sin{\left(2 x \right)} \leq \frac{\sqrt{2}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(2 x \right)} = \frac{\sqrt{2}}{2}$$
Solve:
Given the equation
$$\sin{\left(2 x \right)} = \frac{\sqrt{2}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$2 x = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$2 x = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
Or
$$2 x = 2 \pi n + \frac{\pi}{4}$$
$$2 x = 2 \pi n + \frac{3 \pi}{4}$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
$$x_{1} = \pi n + \frac{\pi}{8}$$
$$x_{2} = \pi n + \frac{3 \pi}{8}$$
$$x_{1} = \pi n + \frac{\pi}{8}$$
$$x_{2} = \pi n + \frac{3 \pi}{8}$$
This roots
$$x_{1} = \pi n + \frac{\pi}{8}$$
$$x_{2} = \pi n + \frac{3 \pi}{8}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \frac{\pi}{8}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \frac{\pi}{8}$$
substitute to the expression
$$\sin{\left(2 x \right)} \leq \frac{\sqrt{2}}{2}$$
$$\sin{\left(2 \left(\pi n - \frac{1}{10} + \frac{\pi}{8}\right) \right)} \leq \frac{\sqrt{2}}{2}$$
___
/ 1 pi \ \/ 2
sin|- - + -- + 2*pi*n| <= -----
\ 5 4 / 2
one of the solutions of our inequality is:
$$x \leq \pi n + \frac{\pi}{8}$$
_____ _____
\ /
-------•-------•-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq \pi n + \frac{\pi}{8}$$
$$x \geq \pi n + \frac{3 \pi}{8}$$
Solving inequality on a graph
Rapid solution 2
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/ ___________\ / ___________\
| / ___ | | / ___ |
|\/ 2 - \/ 2 | |\/ 2 + \/ 2 |
[0, atan|--------------|] U [atan|--------------|, pi]
| ___________| | ___________|
| / ___ | | / ___ |
\\/ 2 + \/ 2 / \\/ 2 - \/ 2 /
$$x\ in\ \left[0, \operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)}\right] \cup \left[\operatorname{atan}{\left(\frac{\sqrt{\sqrt{2} + 2}}{\sqrt{2 - \sqrt{2}}} \right)}, \pi\right]$$
x in Union(Interval(0, atan(sqrt(2 - sqrt(2))/sqrt(sqrt(2) + 2))), Interval(atan(sqrt(sqrt(2) + 2)/sqrt(2 - sqrt(2))), pi))
Rapid solution
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/ / / ___________\\ / / ___________\ \\ | | | / ___ || | | / ___ | || | | |\/ 2 - \/ 2 || | |\/ 2 + \/ 2 | || Or|And|0 <= x, x <= atan|--------------||, And|x <= pi, atan|--------------| <= x|| | | | ___________|| | | ___________| || | | | / ___ || | | / ___ | || \ \ \\/ 2 + \/ 2 // \ \\/ 2 - \/ 2 / //
$$\left(0 \leq x \wedge x \leq \operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)}\right) \vee \left(x \leq \pi \wedge \operatorname{atan}{\left(\frac{\sqrt{\sqrt{2} + 2}}{\sqrt{2 - \sqrt{2}}} \right)} \leq x\right)$$
((0 <= x)∧(x <= atan(sqrt(2 - sqrt(2))/sqrt(2 + sqrt(2)))))∨((x <= pi)∧(atan(sqrt(2 + sqrt(2))/sqrt(2 - sqrt(2))) <= x))