Mister Exam
2(cos4x+sin4x)-3≤0 inequation
The solution
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2*(cos(4*x) + sin(4*x)) - 3 <= 0
$$2 \left(\sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) - 3 \leq 0$$
2*(sin(4*x) + cos(4*x)) - 3 <= 0
Detail solution
Given the inequality:
$$2 \left(\sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) - 3 \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$2 \left(\sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) - 3 = 0$$
Solve:
$$x_{1} = \frac{\operatorname{atan}{\left(\frac{2}{5} - \frac{i}{5} \right)}}{2}$$
$$x_{2} = \frac{\operatorname{atan}{\left(\frac{2}{5} + \frac{i}{5} \right)}}{2}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example
$$-3 + 2 \left(\sin{\left(0 \cdot 4 \right)} + \cos{\left(0 \cdot 4 \right)}\right) \leq 0$$
so the inequality is always executed
$$2 \left(\sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) - 3 \leq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$2 \left(\sin{\left(4 x \right)} + \cos{\left(4 x \right)}\right) - 3 = 0$$
Solve:
$$x_{1} = \frac{\operatorname{atan}{\left(\frac{2}{5} - \frac{i}{5} \right)}}{2}$$
$$x_{2} = \frac{\operatorname{atan}{\left(\frac{2}{5} + \frac{i}{5} \right)}}{2}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example
x0 = 0
$$-3 + 2 \left(\sin{\left(0 \cdot 4 \right)} + \cos{\left(0 \cdot 4 \right)}\right) \leq 0$$
-1 <= 0
so the inequality is always executed
Solving inequality on a graph