Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- log2(1-x)>log2(2x-8)
- y^2-4*x+8>0
- 4x−3x**2−9≤0
- (x-4)^2>16-x^2
- Sum of series:
- sinx
- Graphing y =:
- sinx
- Integral of d{x}:
-
sinx
- Limit of the function:
- -1/4
-
Identical expressions
- sinx<- one / four
- sinus of x less than minus 1 divide by 4
- sinus of x less than minus one divide by four
- sinx<-1 divide by 4
-
Similar expressions
- sin(x)>-√3/2
- (3-(x+5)^-1)/(4*(x+5)^-1-1)<=-0.25
- (3-((x+5)^(-1)))/(4((x+5)^(-1)))<=-0,25
- sinx<+1/4
- sin(x)<=(sqrt(2)/2)
sinx<-1/4 inequation
The solution
Detail solution
Given the inequality:
$$\sin{\left(x \right)} < - \frac{1}{4}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x \right)} = - \frac{1}{4}$$
Solve:
Given the equation
$$\sin{\left(x \right)} = - \frac{1}{4}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{4} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{4} \right)} + \pi$$
Or
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}$$
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} \right)} + \pi$$
, where n - is a integer
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} \right)} + \pi$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} \right)} + \pi$$
This roots
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} \right)} + \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)} - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(x \right)} < - \frac{1}{4}$$
$$\sin{\left(2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)} - \frac{1}{10} \right)} < - \frac{1}{4}$$
one of the solutions of our inequality is:
$$x < 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}$$
$$x > 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} \right)} + \pi$$
$$\sin{\left(x \right)} < - \frac{1}{4}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x \right)} = - \frac{1}{4}$$
Solve:
Given the equation
$$\sin{\left(x \right)} = - \frac{1}{4}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{4} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{4} \right)} + \pi$$
Or
$$x = 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}$$
$$x = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} \right)} + \pi$$
, where n - is a integer
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} \right)} + \pi$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} \right)} + \pi$$
This roots
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} \right)} + \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)} - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(x \right)} < - \frac{1}{4}$$
$$\sin{\left(2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)} - \frac{1}{10} \right)} < - \frac{1}{4}$$
-sin(1/10 - 2*pi*n + asin(1/4)) < -1/4
one of the solutions of our inequality is:
$$x < 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}$$
_____ _____
\ /
-------ο-------ο-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 2 \pi n - \operatorname{asin}{\left(\frac{1}{4} \right)}$$
$$x > 2 \pi n + \operatorname{asin}{\left(\frac{1}{4} \right)} + \pi$$
Solving inequality on a graph
Rapid solution
[src]
/ / ____\ / ____\ \ | |\/ 15 | |\/ 15 | | And|x < - atan|------| + 2*pi, pi + atan|------| < x| \ \ 15 / \ 15 / /
$$x < - \operatorname{atan}{\left(\frac{\sqrt{15}}{15} \right)} + 2 \pi \wedge \operatorname{atan}{\left(\frac{\sqrt{15}}{15} \right)} + \pi < x$$
(pi + atan(sqrt(15)/15) < x)∧(x < -atan(sqrt(15)/15) + 2*pi)
Rapid solution 2
[src]
/ ____\ / ____\
|\/ 15 | |\/ 15 |
(pi + atan|------|, - atan|------| + 2*pi)
\ 15 / \ 15 /
$$x\ in\ \left(\operatorname{atan}{\left(\frac{\sqrt{15}}{15} \right)} + \pi, - \operatorname{atan}{\left(\frac{\sqrt{15}}{15} \right)} + 2 \pi\right)$$
x in Interval.open(atan(sqrt(15)/15) + pi, -atan(sqrt(15)/15) + 2*pi)