Mister Exam
2sin(x)-1>=0 inequation
The solution
Detail solution
Given the inequality:
$$2 \sin{\left(x \right)} - 1 \geq 0$$
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
$$x = 2 \pi n + \frac{\pi}{6}$$
$$x = 2 \pi n + \frac{5 \pi}{6}$$
, where n - is a integer
$$x_{1} = 2 \pi n + \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \frac{5 \pi}{6}$$
$$x_{1} = 2 \pi n + \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \frac{5 \pi}{6}$$
This roots
$$x_{1} = 2 \pi n + \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \frac{5 \pi}{6}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n + \frac{\pi}{6}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10} + \frac{\pi}{6}$$
substitute to the expression
$$2 \sin{\left(x \right)} - 1 \geq 0$$
$$2 \sin{\left(2 \pi n - \frac{1}{10} + \frac{\pi}{6} \right)} - 1 \geq 0$$
but
Then
$$x \leq 2 \pi n + \frac{\pi}{6}$$
no execute
one of the solutions of our inequality is:
$$x \geq 2 \pi n + \frac{\pi}{6} \wedge x \leq 2 \pi n + \frac{5 \pi}{6}$$
$$2 \sin{\left(x \right)} - 1 \geq 0$$
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
$$x = 2 \pi n + \frac{\pi}{6}$$
$$x = 2 \pi n + \frac{5 \pi}{6}$$
, where n - is a integer
$$x_{1} = 2 \pi n + \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \frac{5 \pi}{6}$$
$$x_{1} = 2 \pi n + \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \frac{5 \pi}{6}$$
This roots
$$x_{1} = 2 \pi n + \frac{\pi}{6}$$
$$x_{2} = 2 \pi n + \frac{5 \pi}{6}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n + \frac{\pi}{6}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10} + \frac{\pi}{6}$$
substitute to the expression
$$2 \sin{\left(x \right)} - 1 \geq 0$$
$$2 \sin{\left(2 \pi n - \frac{1}{10} + \frac{\pi}{6} \right)} - 1 \geq 0$$
/ 1 pi \
-1 + 2*sin|- -- + -- + 2*pi*n| >= 0
\ 10 6 / but
/ 1 pi \
-1 + 2*sin|- -- + -- + 2*pi*n| < 0
\ 10 6 / Then
$$x \leq 2 \pi n + \frac{\pi}{6}$$
no execute
one of the solutions of our inequality is:
$$x \geq 2 \pi n + \frac{\pi}{6} \wedge x \leq 2 \pi n + \frac{5 \pi}{6}$$
_____
/ \
-------•-------•-------
x1 x2
Solving inequality on a graph
Rapid solution
[src]
/pi 5*pi\ And|-- <= x, x <= ----| \6 6 /
$$\frac{\pi}{6} \leq x \wedge x \leq \frac{5 \pi}{6}$$
(pi/6 <= x)∧(x <= 5*pi/6)
Rapid solution 2
[src]
pi 5*pi [--, ----] 6 6
$$x\ in\ \left[\frac{\pi}{6}, \frac{5 \pi}{6}\right]$$
x in Interval(pi/6, 5*pi/6)