Mister Exam
3*sin(x/4)<2 inequation
The solution
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/x\
3*sin|-| < 2
\4/
$$3 \sin{\left(\frac{x}{4} \right)} < 2$$
3*sin(x/4) < 2
Detail solution
Given the inequality:
$$3 \sin{\left(\frac{x}{4} \right)} < 2$$
To solve this inequality, we must first solve the corresponding equation:
$$3 \sin{\left(\frac{x}{4} \right)} = 2$$
Solve:
Given the equation
$$3 \sin{\left(\frac{x}{4} \right)} = 2$$
- this is the simplest trigonometric equation
The equation is transformed to
$$\sin{\left(\frac{x}{4} \right)} = \frac{2}{3}$$
This equation is transformed to
$$\frac{x}{4} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{3} \right)}$$
$$\frac{x}{4} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{3} \right)} + \pi$$
Or
$$\frac{x}{4} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{3} \right)}$$
$$\frac{x}{4} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{3} \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{4}$$
$$x_{1} = 8 \pi n + 4 \operatorname{asin}{\left(\frac{2}{3} \right)}$$
$$x_{2} = 8 \pi n - 4 \operatorname{asin}{\left(\frac{2}{3} \right)} + 4 \pi$$
$$x_{1} = 8 \pi n + 4 \operatorname{asin}{\left(\frac{2}{3} \right)}$$
$$x_{2} = 8 \pi n - 4 \operatorname{asin}{\left(\frac{2}{3} \right)} + 4 \pi$$
This roots
$$x_{1} = 8 \pi n + 4 \operatorname{asin}{\left(\frac{2}{3} \right)}$$
$$x_{2} = 8 \pi n - 4 \operatorname{asin}{\left(\frac{2}{3} \right)} + 4 \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(8 \pi n + 4 \operatorname{asin}{\left(\frac{2}{3} \right)}\right) + - \frac{1}{10}$$
=
$$8 \pi n - \frac{1}{10} + 4 \operatorname{asin}{\left(\frac{2}{3} \right)}$$
substitute to the expression
$$3 \sin{\left(\frac{x}{4} \right)} < 2$$
$$3 \sin{\left(\frac{8 \pi n - \frac{1}{10} + 4 \operatorname{asin}{\left(\frac{2}{3} \right)}}{4} \right)} < 2$$
one of the solutions of our inequality is:
$$x < 8 \pi n + 4 \operatorname{asin}{\left(\frac{2}{3} \right)}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 8 \pi n + 4 \operatorname{asin}{\left(\frac{2}{3} \right)}$$
$$x > 8 \pi n - 4 \operatorname{asin}{\left(\frac{2}{3} \right)} + 4 \pi$$
$$3 \sin{\left(\frac{x}{4} \right)} < 2$$
To solve this inequality, we must first solve the corresponding equation:
$$3 \sin{\left(\frac{x}{4} \right)} = 2$$
Solve:
Given the equation
$$3 \sin{\left(\frac{x}{4} \right)} = 2$$
- this is the simplest trigonometric equation
Divide both parts of the equation by 3
The equation is transformed to
$$\sin{\left(\frac{x}{4} \right)} = \frac{2}{3}$$
This equation is transformed to
$$\frac{x}{4} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{3} \right)}$$
$$\frac{x}{4} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{3} \right)} + \pi$$
Or
$$\frac{x}{4} = 2 \pi n + \operatorname{asin}{\left(\frac{2}{3} \right)}$$
$$\frac{x}{4} = 2 \pi n - \operatorname{asin}{\left(\frac{2}{3} \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{4}$$
$$x_{1} = 8 \pi n + 4 \operatorname{asin}{\left(\frac{2}{3} \right)}$$
$$x_{2} = 8 \pi n - 4 \operatorname{asin}{\left(\frac{2}{3} \right)} + 4 \pi$$
$$x_{1} = 8 \pi n + 4 \operatorname{asin}{\left(\frac{2}{3} \right)}$$
$$x_{2} = 8 \pi n - 4 \operatorname{asin}{\left(\frac{2}{3} \right)} + 4 \pi$$
This roots
$$x_{1} = 8 \pi n + 4 \operatorname{asin}{\left(\frac{2}{3} \right)}$$
$$x_{2} = 8 \pi n - 4 \operatorname{asin}{\left(\frac{2}{3} \right)} + 4 \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(8 \pi n + 4 \operatorname{asin}{\left(\frac{2}{3} \right)}\right) + - \frac{1}{10}$$
=
$$8 \pi n - \frac{1}{10} + 4 \operatorname{asin}{\left(\frac{2}{3} \right)}$$
substitute to the expression
$$3 \sin{\left(\frac{x}{4} \right)} < 2$$
$$3 \sin{\left(\frac{8 \pi n - \frac{1}{10} + 4 \operatorname{asin}{\left(\frac{2}{3} \right)}}{4} \right)} < 2$$
3*sin(-1/40 + 2*pi*n + asin(2/3)) < 2
one of the solutions of our inequality is:
$$x < 8 \pi n + 4 \operatorname{asin}{\left(\frac{2}{3} \right)}$$
_____ _____
\ /
-------ο-------ο-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 8 \pi n + 4 \operatorname{asin}{\left(\frac{2}{3} \right)}$$
$$x > 8 \pi n - 4 \operatorname{asin}{\left(\frac{2}{3} \right)} + 4 \pi$$
Solving inequality on a graph
Rapid solution
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/ / / ___\\ / / ___\ \\ | | |3 \/ 5 || | |3 \/ 5 | || Or|And|0 <= x, x < 8*atan|- - -----||, And|x <= 8*pi, 8*atan|- + -----| < x|| \ \ \2 2 // \ \2 2 / //
$$\left(0 \leq x \wedge x < 8 \operatorname{atan}{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)}\right) \vee \left(x \leq 8 \pi \wedge 8 \operatorname{atan}{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)} < x\right)$$
((0 <= x)∧(x < 8*atan(3/2 - sqrt(5)/2)))∨((x <= 8*pi)∧(8*atan(3/2 + sqrt(5)/2) < x))
Rapid solution 2
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/ ___\ / ___\
|3 \/ 5 | |3 \/ 5 |
[0, 8*atan|- - -----|) U (8*atan|- + -----|, 8*pi]
\2 2 / \2 2 /
$$x\ in\ \left[0, 8 \operatorname{atan}{\left(\frac{3}{2} - \frac{\sqrt{5}}{2} \right)}\right) \cup \left(8 \operatorname{atan}{\left(\frac{\sqrt{5}}{2} + \frac{3}{2} \right)}, 8 \pi\right]$$
x in Union(Interval.Ropen(0, 8*atan(3/2 - sqrt(5)/2)), Interval.Lopen(8*atan(sqrt(5)/2 + 3/2), 8*pi))