Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- (1/2)^(2x+1)<4
- (x-3)*(x+7)<=(x+2)^2-7
- 1/cbrt(x*(x-11))>0
- x(3x-4)-10>=4-3x
- Graphing y =:
- 2sin(2x)
- Integral of d{x}:
-
2sin(2x)
- Derivative of:
-
2sin(2x)
-
Identical expressions
- 2sin(2x)<= one
- 2 sinus of (2x) less than or equal to 1
- 2 sinus of (2x) less than or equal to one
- 2sin2x<=1
-
Similar expressions
- 2*sin^2(x)-sqrt(3sinx)-3>0
- (cos^3(x))+1/2*sin^2(x)+1/4*cosx-1/2-1<=0
- (-i*n*(2*x)^4+(2*sin(2*x)^2-cos(x)^2)*cos(x)^2)/(sqrt(2401-7^(x+4)))^2>=0
- -2sin2x<(3)
2sin(2x)<=1 inequation
The solution
Detail solution
Given the inequality:
$$2 \sin{\left(2 x \right)} \leq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$2 \sin{\left(2 x \right)} = 1$$
Solve:
Given the equation
$$2 \sin{\left(2 x \right)} = 1$$
- this is the simplest trigonometric equation
The equation is transformed to
$$\sin{\left(2 x \right)} = \frac{1}{2}$$
This equation is transformed to
$$2 x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$2 x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Or
$$2 x = 2 \pi n + \frac{\pi}{6}$$
$$2 x = 2 \pi n + \frac{5 \pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
$$x_{1} = \pi n + \frac{\pi}{12}$$
$$x_{2} = \pi n + \frac{5 \pi}{12}$$
$$x_{1} = \pi n + \frac{\pi}{12}$$
$$x_{2} = \pi n + \frac{5 \pi}{12}$$
This roots
$$x_{1} = \pi n + \frac{\pi}{12}$$
$$x_{2} = \pi n + \frac{5 \pi}{12}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \frac{\pi}{12}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \frac{\pi}{12}$$
substitute to the expression
$$2 \sin{\left(2 x \right)} \leq 1$$
$$2 \sin{\left(2 \left(\pi n - \frac{1}{10} + \frac{\pi}{12}\right) \right)} \leq 1$$
one of the solutions of our inequality is:
$$x \leq \pi n + \frac{\pi}{12}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq \pi n + \frac{\pi}{12}$$
$$x \geq \pi n + \frac{5 \pi}{12}$$
$$2 \sin{\left(2 x \right)} \leq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$2 \sin{\left(2 x \right)} = 1$$
Solve:
Given the equation
$$2 \sin{\left(2 x \right)} = 1$$
- this is the simplest trigonometric equation
Divide both parts of the equation by 2
The equation is transformed to
$$\sin{\left(2 x \right)} = \frac{1}{2}$$
This equation is transformed to
$$2 x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{2} \right)}$$
$$2 x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{2} \right)} + \pi$$
Or
$$2 x = 2 \pi n + \frac{\pi}{6}$$
$$2 x = 2 \pi n + \frac{5 \pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
$$x_{1} = \pi n + \frac{\pi}{12}$$
$$x_{2} = \pi n + \frac{5 \pi}{12}$$
$$x_{1} = \pi n + \frac{\pi}{12}$$
$$x_{2} = \pi n + \frac{5 \pi}{12}$$
This roots
$$x_{1} = \pi n + \frac{\pi}{12}$$
$$x_{2} = \pi n + \frac{5 \pi}{12}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \frac{\pi}{12}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \frac{\pi}{12}$$
substitute to the expression
$$2 \sin{\left(2 x \right)} \leq 1$$
$$2 \sin{\left(2 \left(\pi n - \frac{1}{10} + \frac{\pi}{12}\right) \right)} \leq 1$$
/ 1 pi \
2*sin|- - + -- + 2*pi*n| <= 1
\ 5 6 / one of the solutions of our inequality is:
$$x \leq \pi n + \frac{\pi}{12}$$
_____ _____
\ /
-------•-------•-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq \pi n + \frac{\pi}{12}$$
$$x \geq \pi n + \frac{5 \pi}{12}$$
Solving inequality on a graph
Rapid solution
[src]
/ / / ___ ___\\ / / ___ ___\ \\ | | |\/ 6 - \/ 2 || | |\/ 2 + \/ 6 | || Or|And|0 <= x, x <= atan|-------------||, And|x <= pi, atan|-------------| <= x|| | | | ___ ___|| | | ___ ___| || \ \ \\/ 2 + \/ 6 // \ \\/ 6 - \/ 2 / //
$$\left(0 \leq x \wedge x \leq \operatorname{atan}{\left(\frac{- \sqrt{2} + \sqrt{6}}{\sqrt{2} + \sqrt{6}} \right)}\right) \vee \left(x \leq \pi \wedge \operatorname{atan}{\left(\frac{\sqrt{2} + \sqrt{6}}{- \sqrt{2} + \sqrt{6}} \right)} \leq x\right)$$
((0 <= x)∧(x <= atan((sqrt(6) - sqrt(2))/(sqrt(2) + sqrt(6)))))∨((x <= pi)∧(atan((sqrt(2) + sqrt(6))/(sqrt(6) - sqrt(2))) <= x))
Rapid solution 2
[src]
/ ___ ___\ / ___ ___\
|\/ 2 - \/ 6 | |\/ 2 + \/ 6 |
[0, -atan|-------------|] U [-atan|-------------|, pi]
| ___ ___| | ___ ___|
\\/ 2 + \/ 6 / \\/ 2 - \/ 6 /
$$x\ in\ \left[0, - \operatorname{atan}{\left(\frac{- \sqrt{6} + \sqrt{2}}{\sqrt{2} + \sqrt{6}} \right)}\right] \cup \left[- \operatorname{atan}{\left(\frac{\sqrt{2} + \sqrt{6}}{- \sqrt{6} + \sqrt{2}} \right)}, \pi\right]$$
x in Union(Interval(0, -atan((-sqrt(6) + sqrt(2))/(sqrt(2) + sqrt(6)))), Interval(-atan((sqrt(2) + sqrt(6))/(-sqrt(6) + sqrt(2))), pi))