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
- -3x+5>-40-3x
- (15^x)-(9*5^x)-(3^x)+9<0
- x-x-2>=0
- Sum of series:
- sinx
- Graphing y =:
- sinx
- Integral of d{x}:
-
sinx
-
Identical expressions
- sinx<= one / six
- sinus of x less than or equal to 1 divide by 6
- sinus of x less than or equal to one divide by six
- sinx<=1 divide by 6
-
Similar expressions
- log(2)*sin((x)/(2))<-1
- sin(x)>((sqrt3)/(2))
sinx<=1/6 inequation
The solution
Detail solution
Given the inequality:
$$\sin{\left(x \right)} \leq \frac{1}{6}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x \right)} = \frac{1}{6}$$
Solve:
Given the equation
$$\sin{\left(x \right)} = \frac{1}{6}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{6} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{6} \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{6} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{6} \right)} + \pi$$
, where n - is a integer
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{6} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{6} \right)} + \pi$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{6} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{6} \right)} + \pi$$
This roots
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{6} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{6} \right)} + \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n + \operatorname{asin}{\left(\frac{1}{6} \right)}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10} + \operatorname{asin}{\left(\frac{1}{6} \right)}$$
substitute to the expression
$$\sin{\left(x \right)} \leq \frac{1}{6}$$
$$\sin{\left(2 \pi n - \frac{1}{10} + \operatorname{asin}{\left(\frac{1}{6} \right)} \right)} \leq \frac{1}{6}$$
one of the solutions of our inequality is:
$$x \leq 2 \pi n + \operatorname{asin}{\left(\frac{1}{6} \right)}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq 2 \pi n + \operatorname{asin}{\left(\frac{1}{6} \right)}$$
$$x \geq 2 \pi n - \operatorname{asin}{\left(\frac{1}{6} \right)} + \pi$$
$$\sin{\left(x \right)} \leq \frac{1}{6}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x \right)} = \frac{1}{6}$$
Solve:
Given the equation
$$\sin{\left(x \right)} = \frac{1}{6}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{6} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{6} \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{6} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{6} \right)} + \pi$$
, where n - is a integer
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{6} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{6} \right)} + \pi$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{6} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{6} \right)} + \pi$$
This roots
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{6} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{6} \right)} + \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n + \operatorname{asin}{\left(\frac{1}{6} \right)}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10} + \operatorname{asin}{\left(\frac{1}{6} \right)}$$
substitute to the expression
$$\sin{\left(x \right)} \leq \frac{1}{6}$$
$$\sin{\left(2 \pi n - \frac{1}{10} + \operatorname{asin}{\left(\frac{1}{6} \right)} \right)} \leq \frac{1}{6}$$
sin(-1/10 + 2*pi*n + asin(1/6)) <= 1/6
one of the solutions of our inequality is:
$$x \leq 2 \pi n + \operatorname{asin}{\left(\frac{1}{6} \right)}$$
_____ _____
\ /
-------•-------•-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq 2 \pi n + \operatorname{asin}{\left(\frac{1}{6} \right)}$$
$$x \geq 2 \pi n - \operatorname{asin}{\left(\frac{1}{6} \right)} + \pi$$
Solving inequality on a graph
Rapid solution
[src]
/ / / ____\\ / / ____\ \\ | | |\/ 35 || | |\/ 35 | || Or|And|0 <= x, x <= atan|------||, And|x <= 2*pi, pi - atan|------| <= x|| \ \ \ 35 // \ \ 35 / //
$$\left(0 \leq x \wedge x \leq \operatorname{atan}{\left(\frac{\sqrt{35}}{35} \right)}\right) \vee \left(x \leq 2 \pi \wedge \pi - \operatorname{atan}{\left(\frac{\sqrt{35}}{35} \right)} \leq x\right)$$
((0 <= x)∧(x <= atan(sqrt(35)/35)))∨((x <= 2*pi)∧(pi - atan(sqrt(35)/35) <= x))
Rapid solution 2
[src]
/ ____\ / ____\
|\/ 35 | |\/ 35 |
[0, atan|------|] U [pi - atan|------|, 2*pi]
\ 35 / \ 35 /
$$x\ in\ \left[0, \operatorname{atan}{\left(\frac{\sqrt{35}}{35} \right)}\right] \cup \left[\pi - \operatorname{atan}{\left(\frac{\sqrt{35}}{35} \right)}, 2 \pi\right]$$
x in Union(Interval(0, atan(sqrt(35)/35)), Interval(pi - atan(sqrt(35)/35), 2*pi))