Mister Exam
sin(2pi+0.5*x)+sin(pi-0.5*x)>=0 inequation
The solution
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/ x\ / x\ sin|2*pi + -| + sin|pi - -| >= 0 \ 2/ \ 2/
$$\sin{\left(\pi - \frac{x}{2} \right)} + \sin{\left(\frac{x}{2} + 2 \pi \right)} \geq 0$$
sin(pi - x/2) + sin(x/2 + 2*pi) >= 0
Detail solution
Given the inequality:
$$\sin{\left(\pi - \frac{x}{2} \right)} + \sin{\left(\frac{x}{2} + 2 \pi \right)} \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(\pi - \frac{x}{2} \right)} + \sin{\left(\frac{x}{2} + 2 \pi \right)} = 0$$
Solve:
Given the equation
$$\sin{\left(\pi - \frac{x}{2} \right)} + \sin{\left(\frac{x}{2} + 2 \pi \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\sin{\left(\pi - \frac{x}{2} \right)} + \sin{\left(\frac{x}{2} + 2 \pi \right)} = 0$$
The equation is transformed to
$$\sin{\left(\frac{x}{2} \right)} = 0$$
This equation is transformed to
$$\frac{x}{2} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$\frac{x}{2} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$\frac{x}{2} = 2 \pi n$$
$$\frac{x}{2} = 2 \pi n + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
$$x_{1} = 4 \pi n$$
$$x_{2} = 4 \pi n + 2 \pi$$
$$x_{1} = 4 \pi n$$
$$x_{2} = 4 \pi n + 2 \pi$$
This roots
$$x_{1} = 4 \pi n$$
$$x_{2} = 4 \pi n + 2 \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$4 \pi n + - \frac{1}{10}$$
=
$$4 \pi n - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(\pi - \frac{x}{2} \right)} + \sin{\left(\frac{x}{2} + 2 \pi \right)} \geq 0$$
$$\sin{\left(\pi - \frac{4 \pi n - \frac{1}{10}}{2} \right)} + \sin{\left(\frac{4 \pi n - \frac{1}{10}}{2} + 2 \pi \right)} \geq 0$$
but
Then
$$x \leq 4 \pi n$$
no execute
one of the solutions of our inequality is:
$$x \geq 4 \pi n \wedge x \leq 4 \pi n + 2 \pi$$
$$\sin{\left(\pi - \frac{x}{2} \right)} + \sin{\left(\frac{x}{2} + 2 \pi \right)} \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(\pi - \frac{x}{2} \right)} + \sin{\left(\frac{x}{2} + 2 \pi \right)} = 0$$
Solve:
Given the equation
$$\sin{\left(\pi - \frac{x}{2} \right)} + \sin{\left(\frac{x}{2} + 2 \pi \right)} = 0$$
- this is the simplest trigonometric equation
with the change of sign in 0
We get:
$$\sin{\left(\pi - \frac{x}{2} \right)} + \sin{\left(\frac{x}{2} + 2 \pi \right)} = 0$$
Divide both parts of the equation by 2
The equation is transformed to
$$\sin{\left(\frac{x}{2} \right)} = 0$$
This equation is transformed to
$$\frac{x}{2} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$\frac{x}{2} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$\frac{x}{2} = 2 \pi n$$
$$\frac{x}{2} = 2 \pi n + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
$$x_{1} = 4 \pi n$$
$$x_{2} = 4 \pi n + 2 \pi$$
$$x_{1} = 4 \pi n$$
$$x_{2} = 4 \pi n + 2 \pi$$
This roots
$$x_{1} = 4 \pi n$$
$$x_{2} = 4 \pi n + 2 \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$4 \pi n + - \frac{1}{10}$$
=
$$4 \pi n - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(\pi - \frac{x}{2} \right)} + \sin{\left(\frac{x}{2} + 2 \pi \right)} \geq 0$$
$$\sin{\left(\pi - \frac{4 \pi n - \frac{1}{10}}{2} \right)} + \sin{\left(\frac{4 \pi n - \frac{1}{10}}{2} + 2 \pi \right)} \geq 0$$
2*sin(-1/20 + 2*pi*n) >= 0
but
2*sin(-1/20 + 2*pi*n) < 0
Then
$$x \leq 4 \pi n$$
no execute
one of the solutions of our inequality is:
$$x \geq 4 \pi n \wedge x \leq 4 \pi n + 2 \pi$$
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Solving inequality on a graph
Rapid solution
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Or(And(0 <= x, x <= 2*pi), x = 4*pi)
$$\left(0 \leq x \wedge x \leq 2 \pi\right) \vee x = 4 \pi$$
(x = 4*pi))∨((0 <= x)∧(x <= 2*pi)
Rapid solution 2
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[0, 2*pi] U {4*pi}
$$x\ in\ \left[0, 2 \pi\right] \cup \left\{4 \pi\right\}$$
x in Union(FiniteSet(4*pi), Interval(0, 2*pi))