Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- (|x|)+(|x-3|)>=3
- 4x-3(5x+8)<_-7
- 4^x<=8^(sqrtx+sqrt1)
- 5x+25<=2x-2
- Graphing y =:
-
sin2x
- Derivative of:
-
sin2x
- Sum of series:
- sin2x
-
Identical expressions
- sin two x=>-(sqrt2/2)
- sinus of 2x equally greater than minus ( square root of 2 divide by 2)
- sinus of two x equally greater than minus ( square root of 2 divide by 2)
- sin2x=>-(√2/2)
- sin2x=>-sqrt2/2
- sin2x=>-(sqrt2 divide by 2)
-
Similar expressions
- sin^2(x)<0,25
- sin2x=>+(sqrt2/2)
- sin(2*x)<cos(x)
- 30*(arcsin^2(x))+23*п*arccos(x)—8,5*(п)^2>=0
- 2*sin(x-pi/4)>=-cos(2*x)-sin(2*x)
sin2x=>-(sqrt2/2) inequation
The solution
You have entered
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___
-\/ 2
sin(2*x) >= -------
2
$$\sin{\left(2 x \right)} \geq - \frac{\sqrt{2}}{2}$$
sin(2*x) >= -sqrt(2)/2
Detail solution
Given the inequality:
$$\sin{\left(2 x \right)} \geq - \frac{\sqrt{2}}{2}$$
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
$$2 x = 2 \pi n - \frac{\pi}{4}$$
$$2 x = 2 \pi n + \frac{5 \pi}{4}$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
$$x_{1} = \pi n - \frac{\pi}{8}$$
$$x_{2} = \pi n + \frac{5 \pi}{8}$$
$$x_{1} = \pi n - \frac{\pi}{8}$$
$$x_{2} = \pi n + \frac{5 \pi}{8}$$
This roots
$$x_{1} = \pi n - \frac{\pi}{8}$$
$$x_{2} = \pi n + \frac{5 \pi}{8}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n - \frac{\pi}{8}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{\pi}{8} - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(2 x \right)} \geq - \frac{\sqrt{2}}{2}$$
$$\sin{\left(2 \left(\pi n - \frac{\pi}{8} - \frac{1}{10}\right) \right)} \geq - \frac{\sqrt{2}}{2}$$
but
Then
$$x \leq \pi n - \frac{\pi}{8}$$
no execute
one of the solutions of our inequality is:
$$x \geq \pi n - \frac{\pi}{8} \wedge x \leq \pi n + \frac{5 \pi}{8}$$
$$\sin{\left(2 x \right)} \geq - \frac{\sqrt{2}}{2}$$
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
$$2 x = 2 \pi n - \frac{\pi}{4}$$
$$2 x = 2 \pi n + \frac{5 \pi}{4}$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
$$x_{1} = \pi n - \frac{\pi}{8}$$
$$x_{2} = \pi n + \frac{5 \pi}{8}$$
$$x_{1} = \pi n - \frac{\pi}{8}$$
$$x_{2} = \pi n + \frac{5 \pi}{8}$$
This roots
$$x_{1} = \pi n - \frac{\pi}{8}$$
$$x_{2} = \pi n + \frac{5 \pi}{8}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n - \frac{\pi}{8}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{\pi}{8} - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(2 x \right)} \geq - \frac{\sqrt{2}}{2}$$
$$\sin{\left(2 \left(\pi n - \frac{\pi}{8} - \frac{1}{10}\right) \right)} \geq - \frac{\sqrt{2}}{2}$$
___
/1 pi \ -\/ 2
-sin|- + -- - 2*pi*n| >= -------
\5 4 / 2
but
___
/1 pi \ -\/ 2
-sin|- + -- - 2*pi*n| < -------
\5 4 / 2
Then
$$x \leq \pi n - \frac{\pi}{8}$$
no execute
one of the solutions of our inequality is:
$$x \geq \pi n - \frac{\pi}{8} \wedge x \leq \pi n + \frac{5 \pi}{8}$$
_____
/ \
-------•-------•-------
x1 x2
Solving inequality on a graph
Rapid solution 2
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/ ___________\ / ___________\
| / ___ | | / ___ |
|\/ 2 + \/ 2 | |\/ 2 - \/ 2 |
[0, pi - atan|--------------|] U [pi - atan|--------------|, pi]
| ___________| | ___________|
| / ___ | | / ___ |
\\/ 2 - \/ 2 / \\/ 2 + \/ 2 /
$$x\ in\ \left[0, \pi - \operatorname{atan}{\left(\frac{\sqrt{\sqrt{2} + 2}}{\sqrt{2 - \sqrt{2}}} \right)}\right] \cup \left[\pi - \operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)}, \pi\right]$$
x in Union(Interval(0, pi - atan(sqrt(sqrt(2) + 2)/sqrt(2 - sqrt(2)))), Interval(pi - atan(sqrt(2 - sqrt(2))/sqrt(sqrt(2) + 2)), pi))
Rapid solution
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/ / / ___________\\ / / ___________\ \\ | | | / ___ || | | / ___ | || | | |\/ 2 + \/ 2 || | |\/ 2 - \/ 2 | || Or|And|0 <= x, x <= pi - atan|--------------||, And|x <= pi, pi - atan|--------------| <= x|| | | | ___________|| | | ___________| || | | | / ___ || | | / ___ | || \ \ \\/ 2 - \/ 2 // \ \\/ 2 + \/ 2 / //
$$\left(0 \leq x \wedge x \leq \pi - \operatorname{atan}{\left(\frac{\sqrt{\sqrt{2} + 2}}{\sqrt{2 - \sqrt{2}}} \right)}\right) \vee \left(x \leq \pi \wedge \pi - \operatorname{atan}{\left(\frac{\sqrt{2 - \sqrt{2}}}{\sqrt{\sqrt{2} + 2}} \right)} \leq x\right)$$
((0 <= x)∧(x <= pi - atan(sqrt(2 + sqrt(2))/sqrt(2 - sqrt(2)))))∨((x <= pi)∧(pi - atan(sqrt(2 - sqrt(2))/sqrt(2 + sqrt(2))) <= x))