Mister Exam
sqrt(3)*sin(2x)+cos(2x)=>1 inequation
The solution
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___ \/ 3 *sin(2*x) + cos(2*x) >= 1
$$\sqrt{3} \sin{\left(2 x \right)} + \cos{\left(2 x \right)} \geq 1$$
sqrt(3)*sin(2*x) + cos(2*x) >= 1
Detail solution
Given the inequality:
$$\sqrt{3} \sin{\left(2 x \right)} + \cos{\left(2 x \right)} \geq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\sqrt{3} \sin{\left(2 x \right)} + \cos{\left(2 x \right)} = 1$$
Solve:
$$x_{1} = 0$$
$$x_{2} = \frac{\pi}{3}$$
$$x_{1} = 0$$
$$x_{2} = \frac{\pi}{3}$$
This roots
$$x_{1} = 0$$
$$x_{2} = \frac{\pi}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10}$$
=
$$- \frac{1}{10}$$
substitute to the expression
$$\sqrt{3} \sin{\left(2 x \right)} + \cos{\left(2 x \right)} \geq 1$$
$$\sqrt{3} \sin{\left(\frac{\left(-1\right) 2}{10} \right)} + \cos{\left(\frac{\left(-1\right) 2}{10} \right)} \geq 1$$
but
Then
$$x \leq 0$$
no execute
one of the solutions of our inequality is:
$$x \geq 0 \wedge x \leq \frac{\pi}{3}$$
$$\sqrt{3} \sin{\left(2 x \right)} + \cos{\left(2 x \right)} \geq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\sqrt{3} \sin{\left(2 x \right)} + \cos{\left(2 x \right)} = 1$$
Solve:
$$x_{1} = 0$$
$$x_{2} = \frac{\pi}{3}$$
$$x_{1} = 0$$
$$x_{2} = \frac{\pi}{3}$$
This roots
$$x_{1} = 0$$
$$x_{2} = \frac{\pi}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10}$$
=
$$- \frac{1}{10}$$
substitute to the expression
$$\sqrt{3} \sin{\left(2 x \right)} + \cos{\left(2 x \right)} \geq 1$$
$$\sqrt{3} \sin{\left(\frac{\left(-1\right) 2}{10} \right)} + \cos{\left(\frac{\left(-1\right) 2}{10} \right)} \geq 1$$
___
- \/ 3 *sin(1/5) + cos(1/5) >= 1
but
___
- \/ 3 *sin(1/5) + cos(1/5) < 1
Then
$$x \leq 0$$
no execute
one of the solutions of our inequality is:
$$x \geq 0 \wedge x \leq \frac{\pi}{3}$$
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Solving inequality on a graph
Rapid solution
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/ / pi\ \ Or|And|0 <= x, x <= --|, x = pi| \ \ 3 / /
$$\left(0 \leq x \wedge x \leq \frac{\pi}{3}\right) \vee x = \pi$$
(x = pi))∨((0 <= x)∧(x <= pi/3)
Rapid solution 2
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pi
[0, --] U {pi}
3
$$x\ in\ \left[0, \frac{\pi}{3}\right] \cup \left\{\pi\right\}$$
x in Union(FiniteSet(pi), Interval(0, pi/3))