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^2-6*x<0-8
- 5^x^2*16^x-1≥5
- 3*x-8<=-5*x+56
- x/(x+3)-(-1+3/x)+13/(x^2+2*x-3)<0
- Derivative of:
-
sin(x)
- Integral of d{x}:
-
sin(x)
- Canonical form:
- sin(x)
-
Identical expressions
- sin(x)>- zero . four
- sinus of (x) greater than minus 0.4
- sinus of (x) greater than minus zero . four
- sinx>-0.4
-
Similar expressions
- sin^2x-3sinxcosx+cos^2x<0
- (sin(x/2)^(2)+3/2*cos(x))*tan(x)<=0
- sin(x)>+0.4
- sinx>-0.4
sin(x)>-0.4 inequation
The solution
Detail solution
Given the inequality:
$$\sin{\left(x \right)} > - \frac{2}{5}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x \right)} = - \frac{2}{5}$$
Solve:
Given the equation
$$\sin{\left(x \right)} = - \frac{2}{5}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(- \frac{2}{5} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(- \frac{2}{5} \right)} + \pi$$
Or
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)}$$
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{2}{5} \right)} + \pi$$
, where n - is a integer
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{5} \right)} + \pi$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{5} \right)} + \pi$$
This roots
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{5} \right)} + \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)} - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(x \right)} > - \frac{2}{5}$$
$$\sin{\left(2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)} - \frac{1}{10} \right)} > - \frac{2}{5}$$
Then
$$x < 2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)}$$
no execute
one of the solutions of our inequality is:
$$x > 2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)} \wedge x < 2 \pi n + \operatorname{asin}{\left(\frac{2}{5} \right)} + \pi$$
$$\sin{\left(x \right)} > - \frac{2}{5}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x \right)} = - \frac{2}{5}$$
Solve:
Given the equation
$$\sin{\left(x \right)} = - \frac{2}{5}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(- \frac{2}{5} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(- \frac{2}{5} \right)} + \pi$$
Or
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)}$$
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{2}{5} \right)} + \pi$$
, where n - is a integer
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{5} \right)} + \pi$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{5} \right)} + \pi$$
This roots
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{5} \right)} + \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)} - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(x \right)} > - \frac{2}{5}$$
$$\sin{\left(2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)} - \frac{1}{10} \right)} > - \frac{2}{5}$$
-sin(1/10 - 2*pi*n + asin(2/5)) > -2/5
Then
$$x < 2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)}$$
no execute
one of the solutions of our inequality is:
$$x > 2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)} \wedge x < 2 \pi n + \operatorname{asin}{\left(\frac{2}{5} \right)} + \pi$$
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x1 x2
Solving inequality on a graph
Rapid solution
[src]
/ / / ____\\ / / ____\ \\ | | |2*\/ 21 || | |2*\/ 21 | || Or|And|0 <= x, x < pi + atan|--------||, And|x <= 2*pi, - atan|--------| + 2*pi < x|| \ \ \ 21 // \ \ 21 / //
$$\left(0 \leq x \wedge x < \operatorname{atan}{\left(\frac{2 \sqrt{21}}{21} \right)} + \pi\right) \vee \left(x \leq 2 \pi \wedge - \operatorname{atan}{\left(\frac{2 \sqrt{21}}{21} \right)} + 2 \pi < x\right)$$
((0 <= x)∧(x < pi + atan(2*sqrt(21)/21)))∨((x <= 2*pi)∧(-atan(2*sqrt(21)/21) + 2*pi < x))
Rapid solution 2
[src]
/ ____\ / ____\
|2*\/ 21 | |2*\/ 21 |
[0, pi + atan|--------|) U (- atan|--------| + 2*pi, 2*pi]
\ 21 / \ 21 /
$$x\ in\ \left[0, \operatorname{atan}{\left(\frac{2 \sqrt{21}}{21} \right)} + \pi\right) \cup \left(- \operatorname{atan}{\left(\frac{2 \sqrt{21}}{21} \right)} + 2 \pi, 2 \pi\right]$$
x in Union(Interval.Ropen(0, atan(2*sqrt(21)/21) + pi), Interval.Lopen(-atan(2*sqrt(21)/21) + 2*pi, 2*pi))