Mister Exam
2+5*sin(5*x)>=0 inequation
The solution
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2 + 5*sin(5*x) >= 0
$$5 \sin{\left(5 x \right)} + 2 \geq 0$$
5*sin(5*x) + 2 >= 0
Detail solution
Given the inequality:
$$5 \sin{\left(5 x \right)} + 2 \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$5 \sin{\left(5 x \right)} + 2 = 0$$
Solve:
Given the equation
$$5 \sin{\left(5 x \right)} + 2 = 0$$
- this is the simplest trigonometric equation
We get:
$$5 \sin{\left(5 x \right)} = -2$$
The equation is transformed to
$$\sin{\left(5 x \right)} = - \frac{2}{5}$$
This equation is transformed to
$$5 x = 2 \pi n + \operatorname{asin}{\left(- \frac{2}{5} \right)}$$
$$5 x = 2 \pi n - \operatorname{asin}{\left(- \frac{2}{5} \right)} + \pi$$
Or
$$5 x = 2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)}$$
$$5 x = 2 \pi n + \operatorname{asin}{\left(\frac{2}{5} \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$5$$
$$x_{1} = \frac{2 \pi n}{5} - \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5}$$
$$x_{2} = \frac{2 \pi n}{5} + \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5} + \frac{\pi}{5}$$
$$x_{1} = \frac{2 \pi n}{5} - \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5}$$
$$x_{2} = \frac{2 \pi n}{5} + \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5} + \frac{\pi}{5}$$
This roots
$$x_{1} = \frac{2 \pi n}{5} - \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5}$$
$$x_{2} = \frac{2 \pi n}{5} + \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5} + \frac{\pi}{5}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{2 \pi n}{5} - \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5}\right) + - \frac{1}{10}$$
=
$$\frac{2 \pi n}{5} - \frac{1}{10} - \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5}$$
substitute to the expression
$$5 \sin{\left(5 x \right)} + 2 \geq 0$$
$$5 \sin{\left(5 \left(\frac{2 \pi n}{5} - \frac{1}{10} - \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5}\right) \right)} + 2 \geq 0$$
but
Then
$$x \leq \frac{2 \pi n}{5} - \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5}$$
no execute
one of the solutions of our inequality is:
$$x \geq \frac{2 \pi n}{5} - \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5} \wedge x \leq \frac{2 \pi n}{5} + \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5} + \frac{\pi}{5}$$
$$5 \sin{\left(5 x \right)} + 2 \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$5 \sin{\left(5 x \right)} + 2 = 0$$
Solve:
Given the equation
$$5 \sin{\left(5 x \right)} + 2 = 0$$
- this is the simplest trigonometric equation
Move 2 to right part of the equation
with the change of sign in 2
We get:
$$5 \sin{\left(5 x \right)} = -2$$
Divide both parts of the equation by 5
The equation is transformed to
$$\sin{\left(5 x \right)} = - \frac{2}{5}$$
This equation is transformed to
$$5 x = 2 \pi n + \operatorname{asin}{\left(- \frac{2}{5} \right)}$$
$$5 x = 2 \pi n - \operatorname{asin}{\left(- \frac{2}{5} \right)} + \pi$$
Or
$$5 x = 2 \pi n - \operatorname{asin}{\left(\frac{2}{5} \right)}$$
$$5 x = 2 \pi n + \operatorname{asin}{\left(\frac{2}{5} \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$5$$
$$x_{1} = \frac{2 \pi n}{5} - \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5}$$
$$x_{2} = \frac{2 \pi n}{5} + \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5} + \frac{\pi}{5}$$
$$x_{1} = \frac{2 \pi n}{5} - \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5}$$
$$x_{2} = \frac{2 \pi n}{5} + \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5} + \frac{\pi}{5}$$
This roots
$$x_{1} = \frac{2 \pi n}{5} - \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5}$$
$$x_{2} = \frac{2 \pi n}{5} + \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5} + \frac{\pi}{5}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{2 \pi n}{5} - \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5}\right) + - \frac{1}{10}$$
=
$$\frac{2 \pi n}{5} - \frac{1}{10} - \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5}$$
substitute to the expression
$$5 \sin{\left(5 x \right)} + 2 \geq 0$$
$$5 \sin{\left(5 \left(\frac{2 \pi n}{5} - \frac{1}{10} - \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5}\right) \right)} + 2 \geq 0$$
2 - 5*sin(1/2 - 2*pi*n + asin(2/5)) >= 0
but
2 - 5*sin(1/2 - 2*pi*n + asin(2/5)) < 0
Then
$$x \leq \frac{2 \pi n}{5} - \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5}$$
no execute
one of the solutions of our inequality is:
$$x \geq \frac{2 \pi n}{5} - \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5} \wedge x \leq \frac{2 \pi n}{5} + \frac{\operatorname{asin}{\left(\frac{2}{5} \right)}}{5} + \frac{\pi}{5}$$
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Solving inequality on a graph
Rapid solution 2
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/ ____\ / ____\
|5 \/ 21 | |5 \/ 21 |
2*atan|- + ------| 2*atan|- - ------|
\2 2 / 2*pi \2 2 / 2*pi 2*pi
[0, - ------------------ + ----] U [- ------------------ + ----, ----]
5 5 5 5 5
$$x\ in\ \left[0, - \frac{2 \operatorname{atan}{\left(\frac{\sqrt{21}}{2} + \frac{5}{2} \right)}}{5} + \frac{2 \pi}{5}\right] \cup \left[- \frac{2 \operatorname{atan}{\left(\frac{5}{2} - \frac{\sqrt{21}}{2} \right)}}{5} + \frac{2 \pi}{5}, \frac{2 \pi}{5}\right]$$
x in Union(Interval(0, -2*atan(sqrt(21)/2 + 5/2)/5 + 2*pi/5), Interval(-2*atan(5/2 - sqrt(21)/2)/5 + 2*pi/5, 2*pi/5))
Rapid solution
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/ / / ____\ \ / / ____\ \\ | | |5 \/ 21 | | | |5 \/ 21 | || | | 2*atan|- + ------| | | 2*atan|- - ------| || | | \2 2 / 2*pi| | 2*pi \2 2 / 2*pi || Or|And|0 <= x, x <= - ------------------ + ----|, And|x <= ----, - ------------------ + ---- <= x|| \ \ 5 5 / \ 5 5 5 //
$$\left(0 \leq x \wedge x \leq - \frac{2 \operatorname{atan}{\left(\frac{\sqrt{21}}{2} + \frac{5}{2} \right)}}{5} + \frac{2 \pi}{5}\right) \vee \left(x \leq \frac{2 \pi}{5} \wedge - \frac{2 \operatorname{atan}{\left(\frac{5}{2} - \frac{\sqrt{21}}{2} \right)}}{5} + \frac{2 \pi}{5} \leq x\right)$$
((0 <= x)∧(x <= -2*atan(5/2 + sqrt(21)/2)/5 + 2*pi/5))∨((x <= 2*pi/5)∧(-2*atan(5/2 - sqrt(21)/2)/5 + 2*pi/5 <= x))