Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- 7-9x>3
- 1+8x<9
- x>1/(x-1)
- 2x-1:5-x>_0
- Sum of series:
-
-1
- Derivative of:
-
-1
- Integral of d{x}:
-
-1
-
Identical expressions
- 7sin*x/ two >- one
- 7 sinus of multiply by x divide by 2 greater than minus 1
- 7 sinus of multiply by x divide by two greater than minus one
- 7sinx/2>-1
- 7sin*x divide by 2>-1
-
Similar expressions
- 7sin(x/2)^2>1
- 7sin(x/2)<-1
- 7sin*x/2>+1
7sin*x/2>-1 inequation
The solution
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[src]
7*sin(x) -------- > -1 2
$$\frac{7 \sin{\left(x \right)}}{2} > -1$$
(7*sin(x))/2 > -1
Detail solution
Given the inequality:
$$\frac{7 \sin{\left(x \right)}}{2} > -1$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{7 \sin{\left(x \right)}}{2} = -1$$
Solve:
Given the equation
$$\frac{7 \sin{\left(x \right)}}{2} = -1$$
- this is the simplest trigonometric equation
The equation is transformed to
$$\sin{\left(x \right)} = - \frac{2}{7}$$
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(- \frac{2}{7} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(- \frac{2}{7} \right)} + \pi$$
Or
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{2}{7} \right)}$$
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{2}{7} \right)} + \pi$$
, where n - is a integer
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{7} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{7} \right)} + \pi$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{7} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{7} \right)} + \pi$$
This roots
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{7} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{7} \right)} + \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n - \operatorname{asin}{\left(\frac{2}{7} \right)}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \operatorname{asin}{\left(\frac{2}{7} \right)} - \frac{1}{10}$$
substitute to the expression
$$\frac{7 \sin{\left(x \right)}}{2} > -1$$
$$\frac{7 \sin{\left(2 \pi n - \operatorname{asin}{\left(\frac{2}{7} \right)} - \frac{1}{10} \right)}}{2} > -1$$
Then
$$x < 2 \pi n - \operatorname{asin}{\left(\frac{2}{7} \right)}$$
no execute
one of the solutions of our inequality is:
$$x > 2 \pi n - \operatorname{asin}{\left(\frac{2}{7} \right)} \wedge x < 2 \pi n + \operatorname{asin}{\left(\frac{2}{7} \right)} + \pi$$
$$\frac{7 \sin{\left(x \right)}}{2} > -1$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{7 \sin{\left(x \right)}}{2} = -1$$
Solve:
Given the equation
$$\frac{7 \sin{\left(x \right)}}{2} = -1$$
- this is the simplest trigonometric equation
Divide both parts of the equation by 7/2
The equation is transformed to
$$\sin{\left(x \right)} = - \frac{2}{7}$$
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(- \frac{2}{7} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(- \frac{2}{7} \right)} + \pi$$
Or
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{2}{7} \right)}$$
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{2}{7} \right)} + \pi$$
, where n - is a integer
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{7} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{7} \right)} + \pi$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{7} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{7} \right)} + \pi$$
This roots
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{7} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{7} \right)} + \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n - \operatorname{asin}{\left(\frac{2}{7} \right)}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \operatorname{asin}{\left(\frac{2}{7} \right)} - \frac{1}{10}$$
substitute to the expression
$$\frac{7 \sin{\left(x \right)}}{2} > -1$$
$$\frac{7 \sin{\left(2 \pi n - \operatorname{asin}{\left(\frac{2}{7} \right)} - \frac{1}{10} \right)}}{2} > -1$$
-7*sin(1/10 - 2*pi*n + asin(2/7))
--------------------------------- > -1
2 Then
$$x < 2 \pi n - \operatorname{asin}{\left(\frac{2}{7} \right)}$$
no execute
one of the solutions of our inequality is:
$$x > 2 \pi n - \operatorname{asin}{\left(\frac{2}{7} \right)} \wedge x < 2 \pi n + \operatorname{asin}{\left(\frac{2}{7} \right)} + \pi$$
_____
/ \
-------ο-------ο-------
x1 x2
Solving inequality on a graph
Rapid solution 2
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/ ___\ / ___\
|2*\/ 5 | |2*\/ 5 |
[0, pi + atan|-------|) U (- atan|-------| + 2*pi, 2*pi]
\ 15 / \ 15 /
$$x\ in\ \left[0, \operatorname{atan}{\left(\frac{2 \sqrt{5}}{15} \right)} + \pi\right) \cup \left(- \operatorname{atan}{\left(\frac{2 \sqrt{5}}{15} \right)} + 2 \pi, 2 \pi\right]$$
x in Union(Interval.Ropen(0, atan(2*sqrt(5)/15) + pi), Interval.Lopen(-atan(2*sqrt(5)/15) + 2*pi, 2*pi))
Rapid solution
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/ / / ___\\ / / ___\ \\ | | |2*\/ 5 || | |2*\/ 5 | || Or|And|0 <= x, x < pi + atan|-------||, And|x <= 2*pi, - atan|-------| + 2*pi < x|| \ \ \ 15 // \ \ 15 / //
$$\left(0 \leq x \wedge x < \operatorname{atan}{\left(\frac{2 \sqrt{5}}{15} \right)} + \pi\right) \vee \left(x \leq 2 \pi \wedge - \operatorname{atan}{\left(\frac{2 \sqrt{5}}{15} \right)} + 2 \pi < x\right)$$
((0 <= x)∧(x < pi + atan(2*sqrt(5)/15)))∨((x <= 2*pi)∧(-atan(2*sqrt(5)/15) + 2*pi < x))