Mister Exam
sin(x/6)>=-2°/2 inequation
The solution
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/-2*pi\
|-----|
/x\ \ 360 /
sin|-| >= -------
\6/ 2
$$\sin{\left(\frac{x}{6} \right)} \geq \frac{\frac{1}{360} \left(- 2 \pi\right)}{2}$$
sin(x/6) >= ((-2*pi)/360)/2
Detail solution
Given the inequality:
$$\sin{\left(\frac{x}{6} \right)} \geq \frac{\frac{1}{360} \left(- 2 \pi\right)}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(\frac{x}{6} \right)} = \frac{\frac{1}{360} \left(- 2 \pi\right)}{2}$$
Solve:
Given the equation
$$\sin{\left(\frac{x}{6} \right)} = \frac{\frac{1}{360} \left(- 2 \pi\right)}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{6} = 2 \pi n + \operatorname{asin}{\left(- \frac{\pi}{360} \right)}$$
$$\frac{x}{6} = 2 \pi n - \operatorname{asin}{\left(- \frac{\pi}{360} \right)} + \pi$$
Or
$$\frac{x}{6} = 2 \pi n - \operatorname{asin}{\left(\frac{\pi}{360} \right)}$$
$$\frac{x}{6} = 2 \pi n + \operatorname{asin}{\left(\frac{\pi}{360} \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{6}$$
$$x_{1} = 12 \pi n - 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)}$$
$$x_{2} = 12 \pi n + 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)} + 6 \pi$$
$$x_{1} = 12 \pi n - 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)}$$
$$x_{2} = 12 \pi n + 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)} + 6 \pi$$
This roots
$$x_{1} = 12 \pi n - 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)}$$
$$x_{2} = 12 \pi n + 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)} + 6 \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(12 \pi n - 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)}\right) + - \frac{1}{10}$$
=
$$12 \pi n - \frac{1}{10} - 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)}$$
substitute to the expression
$$\sin{\left(\frac{x}{6} \right)} \geq \frac{\frac{1}{360} \left(- 2 \pi\right)}{2}$$
$$\sin{\left(\frac{12 \pi n - \frac{1}{10} - 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)}}{6} \right)} \geq \frac{\frac{1}{360} \left(- 2 \pi\right)}{2}$$
but
Then
$$x \leq 12 \pi n - 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)}$$
no execute
one of the solutions of our inequality is:
$$x \geq 12 \pi n - 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)} \wedge x \leq 12 \pi n + 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)} + 6 \pi$$
$$\sin{\left(\frac{x}{6} \right)} \geq \frac{\frac{1}{360} \left(- 2 \pi\right)}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(\frac{x}{6} \right)} = \frac{\frac{1}{360} \left(- 2 \pi\right)}{2}$$
Solve:
Given the equation
$$\sin{\left(\frac{x}{6} \right)} = \frac{\frac{1}{360} \left(- 2 \pi\right)}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{6} = 2 \pi n + \operatorname{asin}{\left(- \frac{\pi}{360} \right)}$$
$$\frac{x}{6} = 2 \pi n - \operatorname{asin}{\left(- \frac{\pi}{360} \right)} + \pi$$
Or
$$\frac{x}{6} = 2 \pi n - \operatorname{asin}{\left(\frac{\pi}{360} \right)}$$
$$\frac{x}{6} = 2 \pi n + \operatorname{asin}{\left(\frac{\pi}{360} \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{6}$$
$$x_{1} = 12 \pi n - 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)}$$
$$x_{2} = 12 \pi n + 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)} + 6 \pi$$
$$x_{1} = 12 \pi n - 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)}$$
$$x_{2} = 12 \pi n + 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)} + 6 \pi$$
This roots
$$x_{1} = 12 \pi n - 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)}$$
$$x_{2} = 12 \pi n + 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)} + 6 \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(12 \pi n - 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)}\right) + - \frac{1}{10}$$
=
$$12 \pi n - \frac{1}{10} - 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)}$$
substitute to the expression
$$\sin{\left(\frac{x}{6} \right)} \geq \frac{\frac{1}{360} \left(- 2 \pi\right)}{2}$$
$$\sin{\left(\frac{12 \pi n - \frac{1}{10} - 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)}}{6} \right)} \geq \frac{\frac{1}{360} \left(- 2 \pi\right)}{2}$$
/1 / pi\\ -pi
-sin|-- - 2*pi*n + asin|---|| >= ----
\60 \360// 360 but
/1 / pi\\ -pi
-sin|-- - 2*pi*n + asin|---|| < ----
\60 \360// 360 Then
$$x \leq 12 \pi n - 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)}$$
no execute
one of the solutions of our inequality is:
$$x \geq 12 \pi n - 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)} \wedge x \leq 12 \pi n + 6 \operatorname{asin}{\left(\frac{\pi}{360} \right)} + 6 \pi$$
_____
/ \
-------•-------•-------
x1 x2
Solving inequality on a graph
Rapid solution
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/ / / __________ __________\ \ / / __________ __________\ \\ | | |360 \/ 360 + pi *\/ 360 - pi | | | |360 \/ 360 + pi *\/ 360 - pi | || Or|And|0 <= x, x <= - 12*atan|--- + -------------------------| + 12*pi|, And|x <= 12*pi, - 12*atan|--- - -------------------------| + 12*pi <= x|| \ \ \ pi pi / / \ \ pi pi / //
$$\left(0 \leq x \wedge x \leq - 12 \operatorname{atan}{\left(\frac{\sqrt{360 - \pi} \sqrt{\pi + 360}}{\pi} + \frac{360}{\pi} \right)} + 12 \pi\right) \vee \left(x \leq 12 \pi \wedge - 12 \operatorname{atan}{\left(- \frac{\sqrt{360 - \pi} \sqrt{\pi + 360}}{\pi} + \frac{360}{\pi} \right)} + 12 \pi \leq x\right)$$
((0 <= x)∧(x <= -12*atan(360/pi + sqrt(360 + pi)*sqrt(360 - pi)/pi) + 12*pi))∨((x <= 12*pi)∧(-12*atan(360/pi - sqrt(360 + pi)*sqrt(360 - pi)/pi) + 12*pi <= x))
Rapid solution 2
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/ __________ __________\ / __________ __________\
|360 \/ 360 + pi *\/ 360 - pi | |360 \/ 360 + pi *\/ 360 - pi |
[0, - 12*atan|--- + -------------------------| + 12*pi] U [- 12*atan|--- - -------------------------| + 12*pi, 12*pi]
\ pi pi / \ pi pi /
$$x\ in\ \left[0, - 12 \operatorname{atan}{\left(\frac{\sqrt{360 - \pi} \sqrt{\pi + 360}}{\pi} + \frac{360}{\pi} \right)} + 12 \pi\right] \cup \left[- 12 \operatorname{atan}{\left(- \frac{\sqrt{360 - \pi} \sqrt{\pi + 360}}{\pi} + \frac{360}{\pi} \right)} + 12 \pi, 12 \pi\right]$$
x in Union(Interval(0, -12*atan(sqrt(360 - pi)*sqrt(pi + 360)/pi + 360/pi) + 12*pi), Interval(-12*atan(-sqrt(360 - pi)*sqrt(pi + 360)/pi + 360/pi) + 12*pi, 12*pi))