Mister Exam
sin(x/4+2)<(-sqrt(2))/2 inequation
The solution
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___ /x \ -\/ 2 sin|- + 2| < ------- \4 / 2
$$\sin{\left(\frac{x}{4} + 2 \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
sin(x/4 + 2) < (-sqrt(2))/2
Detail solution
Given the inequality:
$$\sin{\left(\frac{x}{4} + 2 \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(\frac{x}{4} + 2 \right)} = \frac{\left(-1\right) \sqrt{2}}{2}$$
Solve:
Given the equation
$$\sin{\left(\frac{x}{4} + 2 \right)} = \frac{\left(-1\right) \sqrt{2}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{4} + 2 = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)}$$
$$\frac{x}{4} + 2 = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)} + \pi$$
Or
$$\frac{x}{4} + 2 = 2 \pi n - \frac{\pi}{4}$$
$$\frac{x}{4} + 2 = 2 \pi n + \frac{5 \pi}{4}$$
, where n - is a integer
Move
$$2$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{4} = 2 \pi n - 2 - \frac{\pi}{4}$$
$$\frac{x}{4} = 2 \pi n - 2 + \frac{5 \pi}{4}$$
Divide both parts of the equation by
$$\frac{1}{4}$$
$$x_{1} = 8 \pi n - 8 - \pi$$
$$x_{2} = 8 \pi n - 8 + 5 \pi$$
$$x_{1} = 8 \pi n - 8 - \pi$$
$$x_{2} = 8 \pi n - 8 + 5 \pi$$
This roots
$$x_{1} = 8 \pi n - 8 - \pi$$
$$x_{2} = 8 \pi n - 8 + 5 \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(8 \pi n - 8 - \pi\right) + - \frac{1}{10}$$
=
$$8 \pi n - \frac{81}{10} - \pi$$
substitute to the expression
$$\sin{\left(\frac{x}{4} + 2 \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
$$\sin{\left(\frac{8 \pi n - \frac{81}{10} - \pi}{4} + 2 \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
one of the solutions of our inequality is:
$$x < 8 \pi n - 8 - \pi$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 8 \pi n - 8 - \pi$$
$$x > 8 \pi n - 8 + 5 \pi$$
$$\sin{\left(\frac{x}{4} + 2 \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(\frac{x}{4} + 2 \right)} = \frac{\left(-1\right) \sqrt{2}}{2}$$
Solve:
Given the equation
$$\sin{\left(\frac{x}{4} + 2 \right)} = \frac{\left(-1\right) \sqrt{2}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{4} + 2 = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)}$$
$$\frac{x}{4} + 2 = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)} + \pi$$
Or
$$\frac{x}{4} + 2 = 2 \pi n - \frac{\pi}{4}$$
$$\frac{x}{4} + 2 = 2 \pi n + \frac{5 \pi}{4}$$
, where n - is a integer
Move
$$2$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{4} = 2 \pi n - 2 - \frac{\pi}{4}$$
$$\frac{x}{4} = 2 \pi n - 2 + \frac{5 \pi}{4}$$
Divide both parts of the equation by
$$\frac{1}{4}$$
$$x_{1} = 8 \pi n - 8 - \pi$$
$$x_{2} = 8 \pi n - 8 + 5 \pi$$
$$x_{1} = 8 \pi n - 8 - \pi$$
$$x_{2} = 8 \pi n - 8 + 5 \pi$$
This roots
$$x_{1} = 8 \pi n - 8 - \pi$$
$$x_{2} = 8 \pi n - 8 + 5 \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(8 \pi n - 8 - \pi\right) + - \frac{1}{10}$$
=
$$8 \pi n - \frac{81}{10} - \pi$$
substitute to the expression
$$\sin{\left(\frac{x}{4} + 2 \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
$$\sin{\left(\frac{8 \pi n - \frac{81}{10} - \pi}{4} + 2 \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
___
/1 pi \ -\/ 2
-sin|-- + -- - 2*pi*n| < -------
\40 4 / 2
one of the solutions of our inequality is:
$$x < 8 \pi n - 8 - \pi$$
_____ _____
\ /
-------ο-------ο-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 8 \pi n - 8 - \pi$$
$$x > 8 \pi n - 8 + 5 \pi$$
Solving inequality on a graph
Rapid solution
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/ / 3 ___ ___ 4 ___ / 2 \ \ / 3 ___ ___ 4 ___ / 2 \ \ \ | | - 8*tan (1) + 2*\/ 2 + 8*tan(1) - 2*\/ 2 *tan (1) \/ 2 *\1 + tan (1)/ | |- 8*tan (1) + 2*\/ 2 + 8*tan(1) - 2*\/ 2 *tan (1) \/ 2 *\1 + tan (1)/ | | And|x < 8*pi + 8*atan|- -------------------------------------------------- + --------------------------------|, -8*atan|-------------------------------------------------- + --------------------------------| < x| | | 2 4 ___ ___ 2 | | 2 4 ___ ___ 2 | | \ \ 2 - 12*tan (1) + 2*tan (1) \/ 2 - 4*tan(1) + \/ 2 *tan (1)/ \ 2 - 12*tan (1) + 2*tan (1) \/ 2 - 4*tan(1) + \/ 2 *tan (1)/ /
$$x < 8 \operatorname{atan}{\left(\frac{\sqrt{2} \left(1 + \tan^{2}{\left(1 \right)}\right)}{- 4 \tan{\left(1 \right)} + \sqrt{2} + \sqrt{2} \tan^{2}{\left(1 \right)}} - \frac{- 8 \tan^{3}{\left(1 \right)} - 2 \sqrt{2} \tan^{4}{\left(1 \right)} + 2 \sqrt{2} + 8 \tan{\left(1 \right)}}{- 12 \tan^{2}{\left(1 \right)} + 2 + 2 \tan^{4}{\left(1 \right)}} \right)} + 8 \pi \wedge - 8 \operatorname{atan}{\left(\frac{\sqrt{2} \left(1 + \tan^{2}{\left(1 \right)}\right)}{- 4 \tan{\left(1 \right)} + \sqrt{2} + \sqrt{2} \tan^{2}{\left(1 \right)}} + \frac{- 8 \tan^{3}{\left(1 \right)} - 2 \sqrt{2} \tan^{4}{\left(1 \right)} + 2 \sqrt{2} + 8 \tan{\left(1 \right)}}{- 12 \tan^{2}{\left(1 \right)} + 2 + 2 \tan^{4}{\left(1 \right)}} \right)} < x$$
(-8*atan((-8*tan(1)^3 + 2*sqrt(2) + 8*tan(1) - 2*sqrt(2)*tan(1)^4)/(2 - 12*tan(1)^2 + 2*tan(1)^4) + sqrt(2)*(1 + tan(1)^2)/(sqrt(2) - 4*tan(1) + sqrt(2)*tan(1)^2)) < x)∧(x < 8*pi + 8*atan(-(-8*tan(1)^3 + 2*sqrt(2) + 8*tan(1) - 2*sqrt(2)*tan(1)^4)/(2 - 12*tan(1)^2 + 2*tan(1)^4) + sqrt(2)*(1 + tan(1)^2)/(sqrt(2) - 4*tan(1) + sqrt(2)*tan(1)^2)))
Rapid solution 2
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/ 3 ___ ___ 4 ___ / 2 \ \ / 3 ___ ___ 4 ___ / 2 \ \
|- 8*tan (1) + 2*\/ 2 + 8*tan(1) - 2*\/ 2 *tan (1) \/ 2 *\1 + tan (1)/ | | - 8*tan (1) + 2*\/ 2 + 8*tan(1) - 2*\/ 2 *tan (1) \/ 2 *\1 + tan (1)/ |
(-8*atan|-------------------------------------------------- + --------------------------------|, 8*pi + 8*atan|- -------------------------------------------------- + --------------------------------|)
| 2 4 ___ ___ 2 | | 2 4 ___ ___ 2 |
\ 2 - 12*tan (1) + 2*tan (1) \/ 2 - 4*tan(1) + \/ 2 *tan (1)/ \ 2 - 12*tan (1) + 2*tan (1) \/ 2 - 4*tan(1) + \/ 2 *tan (1)/
$$x\ in\ \left(- 8 \operatorname{atan}{\left(\frac{\sqrt{2} \left(1 + \tan^{2}{\left(1 \right)}\right)}{- 4 \tan{\left(1 \right)} + \sqrt{2} + \sqrt{2} \tan^{2}{\left(1 \right)}} + \frac{- 8 \tan^{3}{\left(1 \right)} - 2 \sqrt{2} \tan^{4}{\left(1 \right)} + 2 \sqrt{2} + 8 \tan{\left(1 \right)}}{- 12 \tan^{2}{\left(1 \right)} + 2 + 2 \tan^{4}{\left(1 \right)}} \right)}, 8 \operatorname{atan}{\left(\frac{\sqrt{2} \left(1 + \tan^{2}{\left(1 \right)}\right)}{- 4 \tan{\left(1 \right)} + \sqrt{2} + \sqrt{2} \tan^{2}{\left(1 \right)}} - \frac{- 8 \tan^{3}{\left(1 \right)} - 2 \sqrt{2} \tan^{4}{\left(1 \right)} + 2 \sqrt{2} + 8 \tan{\left(1 \right)}}{- 12 \tan^{2}{\left(1 \right)} + 2 + 2 \tan^{4}{\left(1 \right)}} \right)} + 8 \pi\right)$$
x in Interval.open(-8*atan(sqrt(2)*(1 + tan(1)^2)/(-4*tan(1) + sqrt(2) + sqrt(2)*tan(1)^2) + (-8*tan(1)^3 - 2*sqrt(2)*tan(1)^4 + 2*sqrt(2) + 8*tan(1))/(-12*tan(1)^2 + 2 + 2*tan(1)^4)), 8*atan(sqrt(2)*(1 + tan(1)^2)/(-4*tan(1) + sqrt(2) + sqrt(2)*tan(1)^2) - (-8*tan(1)^3 - 2*sqrt(2)*tan(1)^4 + 2*sqrt(2) + 8*tan(1))/(-12*tan(1)^2 + 2 + 2*tan(1)^4)) + 8*pi)