Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- 4-9x²<0
- log(1/81)(36-5x)*log(8-x)(1/9)>=1
- logx(3)>0
- 2sinx+1>0
- Derivative of:
-
2sinx+1
- Graphing y =:
- 2sinx+1
-
Identical expressions
- 2sinx+ one > zero
- 2 sinus of x plus 1 greater than 0
- 2 sinus of x plus one greater than zero
-
Similar expressions
- (sinx-2)(sinx+1/2)>=0
- 2sinx-1>0
- 2*sin(x)+1>=0
2sinx+1>0 inequation
The solution
Detail solution
Given the inequality:
$$2 \sin{\left(x \right)} + 1 > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$2 \sin{\left(x \right)} + 1 = 0$$
Solve:
Given the equation
$$2 \sin{\left(x \right)} + 1 = 0$$
- this is the simplest trigonometric equation
We get:
$$2 \sin{\left(x \right)} = -1$$
The equation is transformed to
$$\sin{\left(x \right)} = - \frac{1}{2}$$
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
Or
$$x = 2 \pi n - \frac{\pi}{6}$$
$$x = 2 \pi n + \frac{7 \pi}{6}$$
, where n - is a integer
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \frac{7 \pi}{6}$$
This roots
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \frac{7 \pi}{6}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n - \frac{\pi}{6}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \frac{\pi}{6} - \frac{1}{10}$$
substitute to the expression
$$2 \sin{\left(x \right)} + 1 > 0$$
$$2 \sin{\left(2 \pi n - \frac{\pi}{6} - \frac{1}{10} \right)} + 1 > 0$$
Then
$$x < 2 \pi n - \frac{\pi}{6}$$
no execute
one of the solutions of our inequality is:
$$x > 2 \pi n - \frac{\pi}{6} \wedge x < 2 \pi n + \frac{7 \pi}{6}$$
$$2 \sin{\left(x \right)} + 1 > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$2 \sin{\left(x \right)} + 1 = 0$$
Solve:
Given the equation
$$2 \sin{\left(x \right)} + 1 = 0$$
- this is the simplest trigonometric equation
Move 1 to right part of the equation
with the change of sign in 1
We get:
$$2 \sin{\left(x \right)} = -1$$
Divide both parts of the equation by 2
The equation is transformed to
$$\sin{\left(x \right)} = - \frac{1}{2}$$
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(- \frac{1}{2} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(- \frac{1}{2} \right)} + \pi$$
Or
$$x = 2 \pi n - \frac{\pi}{6}$$
$$x = 2 \pi n + \frac{7 \pi}{6}$$
, where n - is a integer
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \frac{7 \pi}{6}$$
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \frac{7 \pi}{6}$$
This roots
$$x_{1} = 2 \pi n - \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \frac{7 \pi}{6}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n - \frac{\pi}{6}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \frac{\pi}{6} - \frac{1}{10}$$
substitute to the expression
$$2 \sin{\left(x \right)} + 1 > 0$$
$$2 \sin{\left(2 \pi n - \frac{\pi}{6} - \frac{1}{10} \right)} + 1 > 0$$
/1 pi \
1 - 2*sin|-- + -- - 2*pi*n| > 0
\10 6 / Then
$$x < 2 \pi n - \frac{\pi}{6}$$
no execute
one of the solutions of our inequality is:
$$x > 2 \pi n - \frac{\pi}{6} \wedge x < 2 \pi n + \frac{7 \pi}{6}$$
_____
/ \
-------ο-------ο-------
x1 x2
Solving inequality on a graph
Rapid solution 2
[src]
7*pi 11*pi
[0, ----) U (-----, 2*pi]
6 6
$$x\ in\ \left[0, \frac{7 \pi}{6}\right) \cup \left(\frac{11 \pi}{6}, 2 \pi\right]$$
x in Union(Interval.Ropen(0, 7*pi/6), Interval.Lopen(11*pi/6, 2*pi))
Rapid solution
[src]
/ / 7*pi\ / 11*pi \\ Or|And|0 <= x, x < ----|, And|x <= 2*pi, ----- < x|| \ \ 6 / \ 6 //
$$\left(0 \leq x \wedge x < \frac{7 \pi}{6}\right) \vee \left(x \leq 2 \pi \wedge \frac{11 \pi}{6} < x\right)$$
((0 <= x)∧(x < 7*pi/6))∨((x <= 2*pi)∧(11*pi/6 < x))