Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- ab(a+b)+bc(b+c)+ac(a+c)>=6abc
- 2x²-9x+7<0
- (x-2):(4-x)>=0
- log(2-x)<2*log(4)-log(2)
- Limit of the function:
-
sqrt(2)/2
- sqrt(2)/2
-
Identical expressions
- sin(x/ four - three)<sqrt(two)/ two
- sinus of (x divide by 4 minus 3) less than square root of (2) divide by 2
- sinus of (x divide by four minus three) less than square root of (two) divide by two
- sin(x/4-3)<√(2)/2
- sinx/4-3<sqrt2/2
- sin(x divide by 4-3)<sqrt(2) divide by 2
-
Similar expressions
- cos(5x/2)>=-sqrt(2/2)
- sin(x/4+3)<sqrt(2)/2
- cos(3*x-pi/4)<(-sqrt(2))/2
- cos(x)<(-sqrt(2))/2
sin(x/4-3)
The solution
You have entered
[src]
___
/x \ \/ 2
sin|- - 3| < -----
\4 / 2
$$\sin{\left(\frac{x}{4} - 3 \right)} < \frac{\sqrt{2}}{2}$$
sin(x/4 - 3) < sqrt(2)/2
Detail solution
Given the inequality:
$$\sin{\left(\frac{x}{4} - 3 \right)} < \frac{\sqrt{2}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(\frac{x}{4} - 3 \right)} = \frac{\sqrt{2}}{2}$$
Solve:
Given the equation
$$\sin{\left(\frac{x}{4} - 3 \right)} = \frac{\sqrt{2}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{4} - 3 = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$\frac{x}{4} - 3 = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
Or
$$\frac{x}{4} - 3 = 2 \pi n + \frac{\pi}{4}$$
$$\frac{x}{4} - 3 = 2 \pi n + \frac{3 \pi}{4}$$
, where n - is a integer
Move
$$-3$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{4} = 2 \pi n + \frac{\pi}{4} + 3$$
$$\frac{x}{4} = 2 \pi n + \frac{3 \pi}{4} + 3$$
Divide both parts of the equation by
$$\frac{1}{4}$$
$$x_{1} = 8 \pi n + \pi + 12$$
$$x_{2} = 8 \pi n + 3 \pi + 12$$
$$x_{1} = 8 \pi n + \pi + 12$$
$$x_{2} = 8 \pi n + 3 \pi + 12$$
This roots
$$x_{1} = 8 \pi n + \pi + 12$$
$$x_{2} = 8 \pi n + 3 \pi + 12$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(8 \pi n + \pi + 12\right) + - \frac{1}{10}$$
=
$$8 \pi n + \pi + \frac{119}{10}$$
substitute to the expression
$$\sin{\left(\frac{x}{4} - 3 \right)} < \frac{\sqrt{2}}{2}$$
$$\sin{\left(\frac{8 \pi n + \pi + \frac{119}{10}}{4} - 3 \right)} < \frac{\sqrt{2}}{2}$$
___
/ 1 pi \ \/ 2
sin|- -- + -- + 2*pi*n| < -----
\ 40 4 / 2
one of the solutions of our inequality is:
$$x < 8 \pi n + \pi + 12$$
_____ _____
\ /
-------ο-------ο-------
x1 x2
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 8 \pi n + \pi + 12$$
$$x > 8 \pi n + 3 \pi + 12$$
Solving inequality on a graph
Rapid solution
[src]
/ / / ___ 3 ___ 4 ___ / 2 \ \ \ / / ___ 3 ___ 4 ___ / 2 \ \ \\
| | |-8*tan(3/2) - 2*\/ 2 + 8*tan (3/2) + 2*\/ 2 *tan (3/2) \/ 2 *\1 + tan (3/2)/ | | | |-8*tan(3/2) - 2*\/ 2 + 8*tan (3/2) + 2*\/ 2 *tan (3/2) \/ 2 *\1 + tan (3/2)/ | ||
Or|And|0 <= x, x < - 8*atan|------------------------------------------------------- + ------------------------------------| + 8*pi|, And|x <= 8*pi, - 8*atan|------------------------------------------------------- - ------------------------------------| + 8*pi < x||
| | | 2 4 ___ ___ 2 | | | | 2 4 ___ ___ 2 | ||
\ \ \ 2 - 12*tan (3/2) + 2*tan (3/2) \/ 2 - 4*tan(3/2) + \/ 2 *tan (3/2)/ / \ \ 2 - 12*tan (3/2) + 2*tan (3/2) \/ 2 - 4*tan(3/2) + \/ 2 *tan (3/2)/ //
$$\left(0 \leq x \wedge x < - 8 \operatorname{atan}{\left(\frac{\sqrt{2} \left(1 + \tan^{2}{\left(\frac{3}{2} \right)}\right)}{- 4 \tan{\left(\frac{3}{2} \right)} + \sqrt{2} + \sqrt{2} \tan^{2}{\left(\frac{3}{2} \right)}} + \frac{- 8 \tan{\left(\frac{3}{2} \right)} - 2 \sqrt{2} + 8 \tan^{3}{\left(\frac{3}{2} \right)} + 2 \sqrt{2} \tan^{4}{\left(\frac{3}{2} \right)}}{- 12 \tan^{2}{\left(\frac{3}{2} \right)} + 2 + 2 \tan^{4}{\left(\frac{3}{2} \right)}} \right)} + 8 \pi\right) \vee \left(x \leq 8 \pi \wedge - 8 \operatorname{atan}{\left(- \frac{\sqrt{2} \left(1 + \tan^{2}{\left(\frac{3}{2} \right)}\right)}{- 4 \tan{\left(\frac{3}{2} \right)} + \sqrt{2} + \sqrt{2} \tan^{2}{\left(\frac{3}{2} \right)}} + \frac{- 8 \tan{\left(\frac{3}{2} \right)} - 2 \sqrt{2} + 8 \tan^{3}{\left(\frac{3}{2} \right)} + 2 \sqrt{2} \tan^{4}{\left(\frac{3}{2} \right)}}{- 12 \tan^{2}{\left(\frac{3}{2} \right)} + 2 + 2 \tan^{4}{\left(\frac{3}{2} \right)}} \right)} + 8 \pi < x\right)$$
((0 <= x)∧(x < -8*atan((-8*tan(3/2) - 2*sqrt(2) + 8*tan(3/2)^3 + 2*sqrt(2)*tan(3/2)^4)/(2 - 12*tan(3/2)^2 + 2*tan(3/2)^4) + sqrt(2)*(1 + tan(3/2)^2)/(sqrt(2) - 4*tan(3/2) + sqrt(2)*tan(3/2)^2)) + 8*pi))∨((x <= 8*pi)∧(-8*atan((-8*tan(3/2) - 2*sqrt(2) + 8*tan(3/2)^3 + 2*sqrt(2)*tan(3/2)^4)/(2 - 12*tan(3/2)^2 + 2*tan(3/2)^4) - sqrt(2)*(1 + tan(3/2)^2)/(sqrt(2) - 4*tan(3/2) + sqrt(2)*tan(3/2)^2)) + 8*pi < x))
The solution
You have entered
[src]
___ /x \ \/ 2 sin|- - 3| < ----- \4 / 2
$$\sin{\left(\frac{x}{4} - 3 \right)} < \frac{\sqrt{2}}{2}$$
sin(x/4 - 3) < sqrt(2)/2
Detail solution
Given the inequality:
$$\sin{\left(\frac{x}{4} - 3 \right)} < \frac{\sqrt{2}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(\frac{x}{4} - 3 \right)} = \frac{\sqrt{2}}{2}$$
Solve:
Given the equation
$$\sin{\left(\frac{x}{4} - 3 \right)} = \frac{\sqrt{2}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{4} - 3 = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$\frac{x}{4} - 3 = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
Or
$$\frac{x}{4} - 3 = 2 \pi n + \frac{\pi}{4}$$
$$\frac{x}{4} - 3 = 2 \pi n + \frac{3 \pi}{4}$$
, where n - is a integer
Move
$$-3$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{4} = 2 \pi n + \frac{\pi}{4} + 3$$
$$\frac{x}{4} = 2 \pi n + \frac{3 \pi}{4} + 3$$
Divide both parts of the equation by
$$\frac{1}{4}$$
$$x_{1} = 8 \pi n + \pi + 12$$
$$x_{2} = 8 \pi n + 3 \pi + 12$$
$$x_{1} = 8 \pi n + \pi + 12$$
$$x_{2} = 8 \pi n + 3 \pi + 12$$
This roots
$$x_{1} = 8 \pi n + \pi + 12$$
$$x_{2} = 8 \pi n + 3 \pi + 12$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(8 \pi n + \pi + 12\right) + - \frac{1}{10}$$
=
$$8 \pi n + \pi + \frac{119}{10}$$
substitute to the expression
$$\sin{\left(\frac{x}{4} - 3 \right)} < \frac{\sqrt{2}}{2}$$
$$\sin{\left(\frac{8 \pi n + \pi + \frac{119}{10}}{4} - 3 \right)} < \frac{\sqrt{2}}{2}$$
one of the solutions of our inequality is:
$$x < 8 \pi n + \pi + 12$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 8 \pi n + \pi + 12$$
$$x > 8 \pi n + 3 \pi + 12$$
$$\sin{\left(\frac{x}{4} - 3 \right)} < \frac{\sqrt{2}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(\frac{x}{4} - 3 \right)} = \frac{\sqrt{2}}{2}$$
Solve:
Given the equation
$$\sin{\left(\frac{x}{4} - 3 \right)} = \frac{\sqrt{2}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{4} - 3 = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$\frac{x}{4} - 3 = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
Or
$$\frac{x}{4} - 3 = 2 \pi n + \frac{\pi}{4}$$
$$\frac{x}{4} - 3 = 2 \pi n + \frac{3 \pi}{4}$$
, where n - is a integer
Move
$$-3$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{4} = 2 \pi n + \frac{\pi}{4} + 3$$
$$\frac{x}{4} = 2 \pi n + \frac{3 \pi}{4} + 3$$
Divide both parts of the equation by
$$\frac{1}{4}$$
$$x_{1} = 8 \pi n + \pi + 12$$
$$x_{2} = 8 \pi n + 3 \pi + 12$$
$$x_{1} = 8 \pi n + \pi + 12$$
$$x_{2} = 8 \pi n + 3 \pi + 12$$
This roots
$$x_{1} = 8 \pi n + \pi + 12$$
$$x_{2} = 8 \pi n + 3 \pi + 12$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(8 \pi n + \pi + 12\right) + - \frac{1}{10}$$
=
$$8 \pi n + \pi + \frac{119}{10}$$
substitute to the expression
$$\sin{\left(\frac{x}{4} - 3 \right)} < \frac{\sqrt{2}}{2}$$
$$\sin{\left(\frac{8 \pi n + \pi + \frac{119}{10}}{4} - 3 \right)} < \frac{\sqrt{2}}{2}$$
___
/ 1 pi \ \/ 2
sin|- -- + -- + 2*pi*n| < -----
\ 40 4 / 2
one of the solutions of our inequality is:
$$x < 8 \pi n + \pi + 12$$
_____ _____
\ /
-------ο-------ο-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 8 \pi n + \pi + 12$$
$$x > 8 \pi n + 3 \pi + 12$$
Solving inequality on a graph
Rapid solution
[src]
/ / / ___ 3 ___ 4 ___ / 2 \ \ \ / / ___ 3 ___ 4 ___ / 2 \ \ \\ | | |-8*tan(3/2) - 2*\/ 2 + 8*tan (3/2) + 2*\/ 2 *tan (3/2) \/ 2 *\1 + tan (3/2)/ | | | |-8*tan(3/2) - 2*\/ 2 + 8*tan (3/2) + 2*\/ 2 *tan (3/2) \/ 2 *\1 + tan (3/2)/ | || Or|And|0 <= x, x < - 8*atan|------------------------------------------------------- + ------------------------------------| + 8*pi|, And|x <= 8*pi, - 8*atan|------------------------------------------------------- - ------------------------------------| + 8*pi < x|| | | | 2 4 ___ ___ 2 | | | | 2 4 ___ ___ 2 | || \ \ \ 2 - 12*tan (3/2) + 2*tan (3/2) \/ 2 - 4*tan(3/2) + \/ 2 *tan (3/2)/ / \ \ 2 - 12*tan (3/2) + 2*tan (3/2) \/ 2 - 4*tan(3/2) + \/ 2 *tan (3/2)/ //
$$\left(0 \leq x \wedge x < - 8 \operatorname{atan}{\left(\frac{\sqrt{2} \left(1 + \tan^{2}{\left(\frac{3}{2} \right)}\right)}{- 4 \tan{\left(\frac{3}{2} \right)} + \sqrt{2} + \sqrt{2} \tan^{2}{\left(\frac{3}{2} \right)}} + \frac{- 8 \tan{\left(\frac{3}{2} \right)} - 2 \sqrt{2} + 8 \tan^{3}{\left(\frac{3}{2} \right)} + 2 \sqrt{2} \tan^{4}{\left(\frac{3}{2} \right)}}{- 12 \tan^{2}{\left(\frac{3}{2} \right)} + 2 + 2 \tan^{4}{\left(\frac{3}{2} \right)}} \right)} + 8 \pi\right) \vee \left(x \leq 8 \pi \wedge - 8 \operatorname{atan}{\left(- \frac{\sqrt{2} \left(1 + \tan^{2}{\left(\frac{3}{2} \right)}\right)}{- 4 \tan{\left(\frac{3}{2} \right)} + \sqrt{2} + \sqrt{2} \tan^{2}{\left(\frac{3}{2} \right)}} + \frac{- 8 \tan{\left(\frac{3}{2} \right)} - 2 \sqrt{2} + 8 \tan^{3}{\left(\frac{3}{2} \right)} + 2 \sqrt{2} \tan^{4}{\left(\frac{3}{2} \right)}}{- 12 \tan^{2}{\left(\frac{3}{2} \right)} + 2 + 2 \tan^{4}{\left(\frac{3}{2} \right)}} \right)} + 8 \pi < x\right)$$
((0 <= x)∧(x < -8*atan((-8*tan(3/2) - 2*sqrt(2) + 8*tan(3/2)^3 + 2*sqrt(2)*tan(3/2)^4)/(2 - 12*tan(3/2)^2 + 2*tan(3/2)^4) + sqrt(2)*(1 + tan(3/2)^2)/(sqrt(2) - 4*tan(3/2) + sqrt(2)*tan(3/2)^2)) + 8*pi))∨((x <= 8*pi)∧(-8*atan((-8*tan(3/2) - 2*sqrt(2) + 8*tan(3/2)^3 + 2*sqrt(2)*tan(3/2)^4)/(2 - 12*tan(3/2)^2 + 2*tan(3/2)^4) - sqrt(2)*(1 + tan(3/2)^2)/(sqrt(2) - 4*tan(3/2) + sqrt(2)*tan(3/2)^2)) + 8*pi < x))