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):x>1
- asin(x)<=п/2
- 13^(x^2)*22^(x-1)-13<=0
- sinx>=0,2
- Sum of series:
- sinx
- Graphing y =:
- sinx
- Integral of d{x}:
-
sinx
-
Identical expressions
- sinx>= zero , two
- sinus of x greater than or equal to 0,2
- sinus of x greater than or equal to zero , two
- sinx>=O,2
-
Similar expressions
- (sin(x/2))*(cos(x/2))>0
- asin(x)<=п/2
sinx>=0,2 inequation
The solution
Detail solution
Given the inequality:
$$\sin{\left(x \right)} \geq \frac{1}{5}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x \right)} = \frac{1}{5}$$
Solve:
Given the equation
$$\sin{\left(x \right)} = \frac{1}{5}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{5} \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{5} \right)} + \pi$$
, where n - is a integer
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{5} \right)} + \pi$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{5} \right)} + \pi$$
This roots
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{5} \right)} + \pi$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10} + \operatorname{asin}{\left(\frac{1}{5} \right)}$$
substitute to the expression
$$\sin{\left(x \right)} \geq \frac{1}{5}$$
$$\sin{\left(2 \pi n - \frac{1}{10} + \operatorname{asin}{\left(\frac{1}{5} \right)} \right)} \geq \frac{1}{5}$$
but
Then
$$x \leq 2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)}$$
no execute
one of the solutions of our inequality is:
$$x \geq 2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)} \wedge x \leq 2 \pi n - \operatorname{asin}{\left(\frac{1}{5} \right)} + \pi$$
$$\sin{\left(x \right)} \geq \frac{1}{5}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x \right)} = \frac{1}{5}$$
Solve:
Given the equation
$$\sin{\left(x \right)} = \frac{1}{5}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{5} \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{5} \right)} + \pi$$
, where n - is a integer
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{5} \right)} + \pi$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{5} \right)} + \pi$$
This roots
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{5} \right)} + \pi$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10} + \operatorname{asin}{\left(\frac{1}{5} \right)}$$
substitute to the expression
$$\sin{\left(x \right)} \geq \frac{1}{5}$$
$$\sin{\left(2 \pi n - \frac{1}{10} + \operatorname{asin}{\left(\frac{1}{5} \right)} \right)} \geq \frac{1}{5}$$
sin(-1/10 + 2*pi*n + asin(1/5)) >= 1/5
but
sin(-1/10 + 2*pi*n + asin(1/5)) < 1/5
Then
$$x \leq 2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)}$$
no execute
one of the solutions of our inequality is:
$$x \geq 2 \pi n + \operatorname{asin}{\left(\frac{1}{5} \right)} \wedge x \leq 2 \pi n - \operatorname{asin}{\left(\frac{1}{5} \right)} + \pi$$
_____
/ \
-------•-------•-------
x1 x2
Solving inequality on a graph
Rapid solution
[src]
/ / ___\ / ___\ \ | |\/ 6 | |\/ 6 | | And|x <= pi - atan|-----|, atan|-----| <= x| \ \ 12 / \ 12 / /
$$x \leq \pi - \operatorname{atan}{\left(\frac{\sqrt{6}}{12} \right)} \wedge \operatorname{atan}{\left(\frac{\sqrt{6}}{12} \right)} \leq x$$
(atan(sqrt(6)/12) <= x)∧(x <= pi - atan(sqrt(6)/12))
Rapid solution 2
[src]
/ ___\ / ___\
|\/ 6 | |\/ 6 |
[atan|-----|, pi - atan|-----|]
\ 12 / \ 12 /
$$x\ in\ \left[\operatorname{atan}{\left(\frac{\sqrt{6}}{12} \right)}, \pi - \operatorname{atan}{\left(\frac{\sqrt{6}}{12} \right)}\right]$$
x in Interval(atan(sqrt(6)/12), pi - atan(sqrt(6)/12))