Mister Exam
sin(4*x+pi/5)<(-sqrt(3))/2 inequation
The solution
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___ / pi\ -\/ 3 sin|4*x + --| < ------- \ 5 / 2
$$\sin{\left(4 x + \frac{\pi}{5} \right)} < \frac{\left(-1\right) \sqrt{3}}{2}$$
sin(4*x + pi/5) < (-sqrt(3))/2
Detail solution
Given the inequality:
$$\sin{\left(4 x + \frac{\pi}{5} \right)} < \frac{\left(-1\right) \sqrt{3}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(4 x + \frac{\pi}{5} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$
Solve:
Given the equation
$$\sin{\left(4 x + \frac{\pi}{5} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$4 x + \frac{\pi}{5} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)}$$
$$4 x + \frac{\pi}{5} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)} + \pi$$
Or
$$4 x + \frac{\pi}{5} = 2 \pi n - \frac{\pi}{3}$$
$$4 x + \frac{\pi}{5} = 2 \pi n + \frac{4 \pi}{3}$$
, where n - is a integer
Move
$$\frac{\pi}{5}$$
to right part of the equation
with the opposite sign, in total:
$$4 x = 2 \pi n - \frac{8 \pi}{15}$$
$$4 x = 2 \pi n + \frac{17 \pi}{15}$$
Divide both parts of the equation by
$$4$$
$$x_{1} = \frac{\pi n}{2} - \frac{2 \pi}{15}$$
$$x_{2} = \frac{\pi n}{2} + \frac{17 \pi}{60}$$
$$x_{1} = \frac{\pi n}{2} - \frac{2 \pi}{15}$$
$$x_{2} = \frac{\pi n}{2} + \frac{17 \pi}{60}$$
This roots
$$x_{1} = \frac{\pi n}{2} - \frac{2 \pi}{15}$$
$$x_{2} = \frac{\pi n}{2} + \frac{17 \pi}{60}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{\pi n}{2} - \frac{2 \pi}{15}\right) + - \frac{1}{10}$$
=
$$\frac{\pi n}{2} - \frac{2 \pi}{15} - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(4 x + \frac{\pi}{5} \right)} < \frac{\left(-1\right) \sqrt{3}}{2}$$
$$\sin{\left(4 \left(\frac{\pi n}{2} - \frac{2 \pi}{15} - \frac{1}{10}\right) + \frac{\pi}{5} \right)} < \frac{\left(-1\right) \sqrt{3}}{2}$$
one of the solutions of our inequality is:
$$x < \frac{\pi n}{2} - \frac{2 \pi}{15}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < \frac{\pi n}{2} - \frac{2 \pi}{15}$$
$$x > \frac{\pi n}{2} + \frac{17 \pi}{60}$$
$$\sin{\left(4 x + \frac{\pi}{5} \right)} < \frac{\left(-1\right) \sqrt{3}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(4 x + \frac{\pi}{5} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$
Solve:
Given the equation
$$\sin{\left(4 x + \frac{\pi}{5} \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$4 x + \frac{\pi}{5} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)}$$
$$4 x + \frac{\pi}{5} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)} + \pi$$
Or
$$4 x + \frac{\pi}{5} = 2 \pi n - \frac{\pi}{3}$$
$$4 x + \frac{\pi}{5} = 2 \pi n + \frac{4 \pi}{3}$$
, where n - is a integer
Move
$$\frac{\pi}{5}$$
to right part of the equation
with the opposite sign, in total:
$$4 x = 2 \pi n - \frac{8 \pi}{15}$$
$$4 x = 2 \pi n + \frac{17 \pi}{15}$$
Divide both parts of the equation by
$$4$$
$$x_{1} = \frac{\pi n}{2} - \frac{2 \pi}{15}$$
$$x_{2} = \frac{\pi n}{2} + \frac{17 \pi}{60}$$
$$x_{1} = \frac{\pi n}{2} - \frac{2 \pi}{15}$$
$$x_{2} = \frac{\pi n}{2} + \frac{17 \pi}{60}$$
This roots
$$x_{1} = \frac{\pi n}{2} - \frac{2 \pi}{15}$$
$$x_{2} = \frac{\pi n}{2} + \frac{17 \pi}{60}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{\pi n}{2} - \frac{2 \pi}{15}\right) + - \frac{1}{10}$$
=
$$\frac{\pi n}{2} - \frac{2 \pi}{15} - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(4 x + \frac{\pi}{5} \right)} < \frac{\left(-1\right) \sqrt{3}}{2}$$
$$\sin{\left(4 \left(\frac{\pi n}{2} - \frac{2 \pi}{15} - \frac{1}{10}\right) + \frac{\pi}{5} \right)} < \frac{\left(-1\right) \sqrt{3}}{2}$$
___
/2 pi \ -\/ 3
-sin|- + -- - 2*pi*n| < -------
\5 3 / 2
one of the solutions of our inequality is:
$$x < \frac{\pi n}{2} - \frac{2 \pi}{15}$$
_____ _____
\ /
-------ο-------ο-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < \frac{\pi n}{2} - \frac{2 \pi}{15}$$
$$x > \frac{\pi n}{2} + \frac{17 \pi}{60}$$
Solving inequality on a graph
Rapid solution
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/ / ___________ \ / ___________ \ \ | | ___ ____ / ___ | | ___ ____ / ___ | | | | 2*\/ 5 \/ 10 *\/ 3 - \/ 5 | | 2*\/ 5 \/ 10 *\/ 3 - \/ 5 | | | atan|- ------------------------------------------- + -------------------------------------------| atan|------------------------------------------- + -------------------------------------------| | | | _____________ _____________| | _____________ _____________| | | | ____ ___ ___ / ___ ____ ___ ___ / ___ | | ____ ___ ___ / ___ ____ ___ ___ / ___ | | | pi \ \/ 15 - 5*\/ 3 + 2*\/ 5 *\/ 5 - 2*\/ 5 \/ 15 - 5*\/ 3 + 2*\/ 5 *\/ 5 - 2*\/ 5 / pi \\/ 15 - 5*\/ 3 + 2*\/ 5 *\/ 5 - 2*\/ 5 \/ 15 - 5*\/ 3 + 2*\/ 5 *\/ 5 - 2*\/ 5 / | And|x < -- - -------------------------------------------------------------------------------------------------, -- + ----------------------------------------------------------------------------------------------- < x| \ 2 2 2 2 /
$$x < - \frac{\operatorname{atan}{\left(\frac{\sqrt{10} \sqrt{3 - \sqrt{5}}}{- 5 \sqrt{3} + 2 \sqrt{5} \sqrt{5 - 2 \sqrt{5}} + \sqrt{15}} - \frac{2 \sqrt{5}}{- 5 \sqrt{3} + 2 \sqrt{5} \sqrt{5 - 2 \sqrt{5}} + \sqrt{15}} \right)}}{2} + \frac{\pi}{2} \wedge \frac{\operatorname{atan}{\left(\frac{2 \sqrt{5}}{- 5 \sqrt{3} + 2 \sqrt{5} \sqrt{5 - 2 \sqrt{5}} + \sqrt{15}} + \frac{\sqrt{10} \sqrt{3 - \sqrt{5}}}{- 5 \sqrt{3} + 2 \sqrt{5} \sqrt{5 - 2 \sqrt{5}} + \sqrt{15}} \right)}}{2} + \frac{\pi}{2} < x$$
(x < pi/2 - atan(-2*sqrt(5)/(sqrt(15) - 5*sqrt(3) + 2*sqrt(5)*sqrt(5 - 2*sqrt(5))) + sqrt(10)*sqrt(3 - sqrt(5))/(sqrt(15) - 5*sqrt(3) + 2*sqrt(5)*sqrt(5 - 2*sqrt(5))))/2)∧(pi/2 + atan(2*sqrt(5)/(sqrt(15) - 5*sqrt(3) + 2*sqrt(5)*sqrt(5 - 2*sqrt(5))) + sqrt(10)*sqrt(3 - sqrt(5))/(sqrt(15) - 5*sqrt(3) + 2*sqrt(5)*sqrt(5 - 2*sqrt(5))))/2 < x)
Rapid solution 2
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/ ___________ \ / ___________ \
| ___ ____ / ___ | | ___ ____ / ___ |
| 2*\/ 5 \/ 10 *\/ 3 - \/ 5 | | 2*\/ 5 \/ 10 *\/ 3 - \/ 5 |
atan|------------------------------------------- + -------------------------------------------| atan|- ------------------------------------------- + -------------------------------------------|
| _____________ _____________| | _____________ _____________|
| ____ ___ ___ / ___ ____ ___ ___ / ___ | | ____ ___ ___ / ___ ____ ___ ___ / ___ |
pi \\/ 15 - 5*\/ 3 + 2*\/ 5 *\/ 5 - 2*\/ 5 \/ 15 - 5*\/ 3 + 2*\/ 5 *\/ 5 - 2*\/ 5 / pi \ \/ 15 - 5*\/ 3 + 2*\/ 5 *\/ 5 - 2*\/ 5 \/ 15 - 5*\/ 3 + 2*\/ 5 *\/ 5 - 2*\/ 5 /
(-- + -----------------------------------------------------------------------------------------------, -- - -------------------------------------------------------------------------------------------------)
2 2 2 2
$$x\ in\ \left(\frac{\operatorname{atan}{\left(\frac{2 \sqrt{5}}{- 5 \sqrt{3} + 2 \sqrt{5} \sqrt{5 - 2 \sqrt{5}} + \sqrt{15}} + \frac{\sqrt{10} \sqrt{3 - \sqrt{5}}}{- 5 \sqrt{3} + 2 \sqrt{5} \sqrt{5 - 2 \sqrt{5}} + \sqrt{15}} \right)}}{2} + \frac{\pi}{2}, - \frac{\operatorname{atan}{\left(\frac{\sqrt{10} \sqrt{3 - \sqrt{5}}}{- 5 \sqrt{3} + 2 \sqrt{5} \sqrt{5 - 2 \sqrt{5}} + \sqrt{15}} - \frac{2 \sqrt{5}}{- 5 \sqrt{3} + 2 \sqrt{5} \sqrt{5 - 2 \sqrt{5}} + \sqrt{15}} \right)}}{2} + \frac{\pi}{2}\right)$$
x in Interval.open(atan(2*sqrt(5)/(-5*sqrt(3) + 2*sqrt(5)*sqrt(5 - 2*sqrt(5)) + sqrt(15)) + sqrt(10)*sqrt(3 - sqrt(5))/(-5*sqrt(3) + 2*sqrt(5)*sqrt(5 - 2*sqrt(5)) + sqrt(15)))/2 + pi/2, -atan(sqrt(10)*sqrt(3 - sqrt(5))/(-5*sqrt(3) + 2*sqrt(5)*sqrt(5 - 2*sqrt(5)) + sqrt(15)) - 2*sqrt(5)/(-5*sqrt(3) + 2*sqrt(5)*sqrt(5 - 2*sqrt(5)) + sqrt(15)))/2 + pi/2)