Mister Exam
sin2x*cos2x≥1/4 inequation
The solution
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sin(2*x)*cos(2*x) >= 1/4
$$\sin{\left(2 x \right)} \cos{\left(2 x \right)} \geq \frac{1}{4}$$
sin(2*x)*cos(2*x) >= 1/4
Detail solution
Given the inequality:
$$\sin{\left(2 x \right)} \cos{\left(2 x \right)} \geq \frac{1}{4}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(2 x \right)} \cos{\left(2 x \right)} = \frac{1}{4}$$
Solve:
$$x_{1} = \operatorname{atan}{\left(-2 + 2 \sqrt{2 - \sqrt{3}} + \sqrt{3} \right)}$$
$$x_{2} = - \operatorname{atan}{\left(- \sqrt{3} + 2 \sqrt{2 - \sqrt{3}} + 2 \right)}$$
$$x_{3} = - \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)}$$
$$x_{4} = - \operatorname{atan}{\left(- 2 \sqrt{\sqrt{3} + 2} + \sqrt{3} + 2 \right)}$$
$$x_{1} = \operatorname{atan}{\left(-2 + 2 \sqrt{2 - \sqrt{3}} + \sqrt{3} \right)}$$
$$x_{2} = - \operatorname{atan}{\left(- \sqrt{3} + 2 \sqrt{2 - \sqrt{3}} + 2 \right)}$$
$$x_{3} = - \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)}$$
$$x_{4} = - \operatorname{atan}{\left(- 2 \sqrt{\sqrt{3} + 2} + \sqrt{3} + 2 \right)}$$
This roots
$$x_{3} = - \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)}$$
$$x_{2} = - \operatorname{atan}{\left(- \sqrt{3} + 2 \sqrt{2 - \sqrt{3}} + 2 \right)}$$
$$x_{4} = - \operatorname{atan}{\left(- 2 \sqrt{\sqrt{3} + 2} + \sqrt{3} + 2 \right)}$$
$$x_{1} = \operatorname{atan}{\left(-2 + 2 \sqrt{2 - \sqrt{3}} + \sqrt{3} \right)}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{3}$$
For example, let's take the point
$$x_{0} = x_{3} - \frac{1}{10}$$
=
$$- \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)} - \frac{1}{10}$$
=
$$- \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)} - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(2 x \right)} \cos{\left(2 x \right)} \geq \frac{1}{4}$$
$$\sin{\left(2 \left(- \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)} - \frac{1}{10}\right) \right)} \cos{\left(2 \left(- \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)} - \frac{1}{10}\right) \right)} \geq \frac{1}{4}$$
but
Then
$$x \leq - \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)}$$
no execute
one of the solutions of our inequality is:
$$x \geq - \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)} \wedge x \leq - \operatorname{atan}{\left(- \sqrt{3} + 2 \sqrt{2 - \sqrt{3}} + 2 \right)}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \geq - \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)} \wedge x \leq - \operatorname{atan}{\left(- \sqrt{3} + 2 \sqrt{2 - \sqrt{3}} + 2 \right)}$$
$$x \geq - \operatorname{atan}{\left(- 2 \sqrt{\sqrt{3} + 2} + \sqrt{3} + 2 \right)} \wedge x \leq \operatorname{atan}{\left(-2 + 2 \sqrt{2 - \sqrt{3}} + \sqrt{3} \right)}$$
$$\sin{\left(2 x \right)} \cos{\left(2 x \right)} \geq \frac{1}{4}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(2 x \right)} \cos{\left(2 x \right)} = \frac{1}{4}$$
Solve:
$$x_{1} = \operatorname{atan}{\left(-2 + 2 \sqrt{2 - \sqrt{3}} + \sqrt{3} \right)}$$
$$x_{2} = - \operatorname{atan}{\left(- \sqrt{3} + 2 \sqrt{2 - \sqrt{3}} + 2 \right)}$$
$$x_{3} = - \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)}$$
$$x_{4} = - \operatorname{atan}{\left(- 2 \sqrt{\sqrt{3} + 2} + \sqrt{3} + 2 \right)}$$
$$x_{1} = \operatorname{atan}{\left(-2 + 2 \sqrt{2 - \sqrt{3}} + \sqrt{3} \right)}$$
$$x_{2} = - \operatorname{atan}{\left(- \sqrt{3} + 2 \sqrt{2 - \sqrt{3}} + 2 \right)}$$
$$x_{3} = - \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)}$$
$$x_{4} = - \operatorname{atan}{\left(- 2 \sqrt{\sqrt{3} + 2} + \sqrt{3} + 2 \right)}$$
This roots
$$x_{3} = - \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)}$$
$$x_{2} = - \operatorname{atan}{\left(- \sqrt{3} + 2 \sqrt{2 - \sqrt{3}} + 2 \right)}$$
$$x_{4} = - \operatorname{atan}{\left(- 2 \sqrt{\sqrt{3} + 2} + \sqrt{3} + 2 \right)}$$
$$x_{1} = \operatorname{atan}{\left(-2 + 2 \sqrt{2 - \sqrt{3}} + \sqrt{3} \right)}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{3}$$
For example, let's take the point
$$x_{0} = x_{3} - \frac{1}{10}$$
=
$$- \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)} - \frac{1}{10}$$
=
$$- \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)} - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(2 x \right)} \cos{\left(2 x \right)} \geq \frac{1}{4}$$
$$\sin{\left(2 \left(- \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)} - \frac{1}{10}\right) \right)} \cos{\left(2 \left(- \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)} - \frac{1}{10}\right) \right)} \geq \frac{1}{4}$$
/ / ___________\\ / / ___________\\
|1 | ___ / ___ || |1 | ___ / ___ ||
-cos|- + 2*atan\2 + \/ 3 + 2*\/ 2 + \/ 3 /|*sin|- + 2*atan\2 + \/ 3 + 2*\/ 2 + \/ 3 /| >= 1/4
\5 / \5 /
but
/ / ___________\\ / / ___________\\
|1 | ___ / ___ || |1 | ___ / ___ ||
-cos|- + 2*atan\2 + \/ 3 + 2*\/ 2 + \/ 3 /|*sin|- + 2*atan\2 + \/ 3 + 2*\/ 2 + \/ 3 /| < 1/4
\5 / \5 /
Then
$$x \leq - \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)}$$
no execute
one of the solutions of our inequality is:
$$x \geq - \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)} \wedge x \leq - \operatorname{atan}{\left(- \sqrt{3} + 2 \sqrt{2 - \sqrt{3}} + 2 \right)}$$
_____ _____
/ \ / \
-------•-------•-------•-------•-------
x3 x2 x4 x1Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \geq - \operatorname{atan}{\left(\sqrt{3} + 2 + 2 \sqrt{\sqrt{3} + 2} \right)} \wedge x \leq - \operatorname{atan}{\left(- \sqrt{3} + 2 \sqrt{2 - \sqrt{3}} + 2 \right)}$$
$$x \geq - \operatorname{atan}{\left(- 2 \sqrt{\sqrt{3} + 2} + \sqrt{3} + 2 \right)} \wedge x \leq \operatorname{atan}{\left(-2 + 2 \sqrt{2 - \sqrt{3}} + \sqrt{3} \right)}$$
Solving inequality on a graph
Rapid solution
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/pi 5*pi\ And|-- <= x, x <= ----| \24 24 /
$$\frac{\pi}{24} \leq x \wedge x \leq \frac{5 \pi}{24}$$
(pi/24 <= x)∧(x <= 5*pi/24)
Rapid solution 2
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pi 5*pi [--, ----] 24 24
$$x\ in\ \left[\frac{\pi}{24}, \frac{5 \pi}{24}\right]$$
x in Interval(pi/24, 5*pi/24)