Mister Exam
cot(1/2*x-pi/4)<=-sqrt(3)/3 inequation
The solution
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[src]
___ /x pi\ -\/ 3 cot|- - --| <= ------- \2 4 / 3
$$\cot{\left(\frac{x}{2} - \frac{\pi}{4} \right)} \leq \frac{\left(-1\right) \sqrt{3}}{3}$$
cot(x/2 - pi/4) <= (-sqrt(3))/3
Detail solution
Given the inequality:
$$\cot{\left(\frac{x}{2} - \frac{\pi}{4} \right)} \leq \frac{\left(-1\right) \sqrt{3}}{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cot{\left(\frac{x}{2} - \frac{\pi}{4} \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
Solve:
$$x_{1} = - \frac{\pi}{6}$$
$$x_{1} = - \frac{\pi}{6}$$
This roots
$$x_{1} = - \frac{\pi}{6}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{\pi}{6} - \frac{1}{10}$$
=
$$- \frac{\pi}{6} - \frac{1}{10}$$
substitute to the expression
$$\cot{\left(\frac{x}{2} - \frac{\pi}{4} \right)} \leq \frac{\left(-1\right) \sqrt{3}}{3}$$
$$\cot{\left(- \frac{\pi}{4} + \frac{- \frac{\pi}{6} - \frac{1}{10}}{2} \right)} \leq \frac{\left(-1\right) \sqrt{3}}{3}$$
but
Then
$$x \leq - \frac{\pi}{6}$$
no execute
the solution of our inequality is:
$$x \geq - \frac{\pi}{6}$$
$$\cot{\left(\frac{x}{2} - \frac{\pi}{4} \right)} \leq \frac{\left(-1\right) \sqrt{3}}{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cot{\left(\frac{x}{2} - \frac{\pi}{4} \right)} = \frac{\left(-1\right) \sqrt{3}}{3}$$
Solve:
$$x_{1} = - \frac{\pi}{6}$$
$$x_{1} = - \frac{\pi}{6}$$
This roots
$$x_{1} = - \frac{\pi}{6}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{\pi}{6} - \frac{1}{10}$$
=
$$- \frac{\pi}{6} - \frac{1}{10}$$
substitute to the expression
$$\cot{\left(\frac{x}{2} - \frac{\pi}{4} \right)} \leq \frac{\left(-1\right) \sqrt{3}}{3}$$
$$\cot{\left(- \frac{\pi}{4} + \frac{- \frac{\pi}{6} - \frac{1}{10}}{2} \right)} \leq \frac{\left(-1\right) \sqrt{3}}{3}$$
___
/1 pi\ -\/ 3
-cot|-- + --| <= -------
\20 3 / 3
but
___
/1 pi\ -\/ 3
-cot|-- + --| >= -------
\20 3 / 3
Then
$$x \leq - \frac{\pi}{6}$$
no execute
the solution of our inequality is:
$$x \geq - \frac{\pi}{6}$$
_____
/
-------•-------
x1
Rapid solution
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/ / / ___ \ \\ | / pi\ | | ___ 2*\/ 6 | || Or|And|0 <= x, x < --|, And|x <= 2*pi, 4*atan|2 + \/ 3 + ---------| <= x|| | \ 2 / | | ___| || \ \ \ 3 - \/ 3 / //
$$\left(0 \leq x \wedge x < \frac{\pi}{2}\right) \vee \left(x \leq 2 \pi \wedge 4 \operatorname{atan}{\left(\sqrt{3} + 2 + \frac{2 \sqrt{6}}{3 - \sqrt{3}} \right)} \leq x\right)$$
((0 <= x)∧(x < pi/2))∨((x <= 2*pi)∧(4*atan(2 + sqrt(3) + 2*sqrt(6)/(3 - sqrt(3))) <= x))
Rapid solution 2
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/ ___ \
pi | ___ 2*\/ 6 |
[0, --) U [4*atan|2 + \/ 3 + ---------|, 2*pi]
2 | ___|
\ 3 - \/ 3 /
$$x\ in\ \left[0, \frac{\pi}{2}\right) \cup \left[4 \operatorname{atan}{\left(\sqrt{3} + 2 + \frac{2 \sqrt{6}}{3 - \sqrt{3}} \right)}, 2 \pi\right]$$
x in Union(Interval.Ropen(0, pi/2), Interval(4*atan(sqrt(3) + 2 + 2*sqrt(6)/(3 - sqrt(3))), 2*pi))