Mister Exam
sin(4x/3)>-√3/2 inequation
The solution
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___ /4*x\ -\/ 3 sin|---| > ------- \ 3 / 2
$$\sin{\left(\frac{4 x}{3} \right)} > \frac{\left(-1\right) \sqrt{3}}{2}$$
sin((4*x)/3) > (-sqrt(3))/2
Detail solution
Given the inequality:
$$\sin{\left(\frac{4 x}{3} \right)} > \frac{\left(-1\right) \sqrt{3}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(\frac{4 x}{3} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$
Solve:
Given the equation
$$\sin{\left(\frac{4 x}{3} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{4 x}{3} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)}$$
$$\frac{4 x}{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)} + \pi$$
Or
$$\frac{4 x}{3} = 2 \pi n - \frac{\pi}{3}$$
$$\frac{4 x}{3} = 2 \pi n + \frac{4 \pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{4}{3}$$
$$x_{1} = \frac{3 \pi n}{2} - \frac{\pi}{4}$$
$$x_{2} = \frac{3 \pi n}{2} + \pi$$
$$x_{1} = \frac{3 \pi n}{2} - \frac{\pi}{4}$$
$$x_{2} = \frac{3 \pi n}{2} + \pi$$
This roots
$$x_{1} = \frac{3 \pi n}{2} - \frac{\pi}{4}$$
$$x_{2} = \frac{3 \pi n}{2} + \pi$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{3 \pi n}{2} - \frac{\pi}{4}\right) + - \frac{1}{10}$$
=
$$\frac{3 \pi n}{2} - \frac{\pi}{4} - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(\frac{4 x}{3} \right)} > \frac{\left(-1\right) \sqrt{3}}{2}$$
$$\sin{\left(\frac{4 \left(\frac{3 \pi n}{2} - \frac{\pi}{4} - \frac{1}{10}\right)}{3} \right)} > \frac{\left(-1\right) \sqrt{3}}{2}$$
Then
$$x < \frac{3 \pi n}{2} - \frac{\pi}{4}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{3 \pi n}{2} - \frac{\pi}{4} \wedge x < \frac{3 \pi n}{2} + \pi$$
$$\sin{\left(\frac{4 x}{3} \right)} > \frac{\left(-1\right) \sqrt{3}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(\frac{4 x}{3} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$
Solve:
Given the equation
$$\sin{\left(\frac{4 x}{3} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{4 x}{3} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)}$$
$$\frac{4 x}{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)} + \pi$$
Or
$$\frac{4 x}{3} = 2 \pi n - \frac{\pi}{3}$$
$$\frac{4 x}{3} = 2 \pi n + \frac{4 \pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{4}{3}$$
$$x_{1} = \frac{3 \pi n}{2} - \frac{\pi}{4}$$
$$x_{2} = \frac{3 \pi n}{2} + \pi$$
$$x_{1} = \frac{3 \pi n}{2} - \frac{\pi}{4}$$
$$x_{2} = \frac{3 \pi n}{2} + \pi$$
This roots
$$x_{1} = \frac{3 \pi n}{2} - \frac{\pi}{4}$$
$$x_{2} = \frac{3 \pi n}{2} + \pi$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{3 \pi n}{2} - \frac{\pi}{4}\right) + - \frac{1}{10}$$
=
$$\frac{3 \pi n}{2} - \frac{\pi}{4} - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(\frac{4 x}{3} \right)} > \frac{\left(-1\right) \sqrt{3}}{2}$$
$$\sin{\left(\frac{4 \left(\frac{3 \pi n}{2} - \frac{\pi}{4} - \frac{1}{10}\right)}{3} \right)} > \frac{\left(-1\right) \sqrt{3}}{2}$$
___
/2 pi \ -\/ 3
-sin|-- + -- - 2*pi*n| > -------
\15 3 / 2
Then
$$x < \frac{3 \pi n}{2} - \frac{\pi}{4}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{3 \pi n}{2} - \frac{\pi}{4} \wedge x < \frac{3 \pi n}{2} + \pi$$
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/ \
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x1 x2
Solving inequality on a graph
Rapid solution 2
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5*pi 3*pi
[0, pi) U (----, ----]
4 2
$$x\ in\ \left[0, \pi\right) \cup \left(\frac{5 \pi}{4}, \frac{3 \pi}{2}\right]$$
x in Union(Interval.Ropen(0, pi), Interval.Lopen(5*pi/4, 3*pi/2))
Rapid solution
[src]
/ / 3*pi 5*pi \\ Or|And(0 <= x, x < pi), And|x <= ----, ---- < x|| \ \ 2 4 //
$$\left(0 \leq x \wedge x < \pi\right) \vee \left(x \leq \frac{3 \pi}{2} \wedge \frac{5 \pi}{4} < x\right)$$
((0 <= x)∧(x < pi))∨((x <= 3*pi/2)∧(5*pi/4 < x))