Mister Exam
2\(cosx)>0 inequation
The solution
Detail solution
Given the inequality:
$$\frac{2}{\cos{\left(x \right)}} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{2}{\cos{\left(x \right)}} = 0$$
Solve:
Given the equation
$$\frac{2}{\cos{\left(x \right)}} = 0$$
transform
$$\frac{2}{\cos{\left(x \right)}} = 0$$
$$\frac{2}{\cos{\left(x \right)}} = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Given the equation:
$$\frac{2}{w} = 0$$
Multiply the equation sides by the denominator w
we get:
Move free summands (without w)
from left part to right part, we given:
$$0 = -2$$
This equation has no roots
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
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
$$\frac{2}{\cos{\left(0 \right)}} > 0$$
so the inequality is always executed
$$\frac{2}{\cos{\left(x \right)}} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\frac{2}{\cos{\left(x \right)}} = 0$$
Solve:
Given the equation
$$\frac{2}{\cos{\left(x \right)}} = 0$$
transform
$$\frac{2}{\cos{\left(x \right)}} = 0$$
$$\frac{2}{\cos{\left(x \right)}} = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
Given the equation:
$$\frac{2}{w} = 0$$
Multiply the equation sides by the denominator w
we get:
False
Move free summands (without w)
from left part to right part, we given:
$$0 = -2$$
This equation has no roots
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
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
$$\frac{2}{\cos{\left(0 \right)}} > 0$$
2 > 0
so the inequality is always executed
Solving inequality on a graph
Rapid solution
[src]
/ / pi\ / 3*pi \\ Or|And|0 <= x, x < --|, And|x <= 2*pi, ---- < x|| \ \ 2 / \ 2 //
$$\left(0 \leq x \wedge x < \frac{\pi}{2}\right) \vee \left(x \leq 2 \pi \wedge \frac{3 \pi}{2} < x\right)$$
((0 <= x)∧(x < pi/2))∨((x <= 2*pi)∧(3*pi/2 < x))
Rapid solution 2
[src]
pi 3*pi
[0, --) U (----, 2*pi]
2 2
$$x\ in\ \left[0, \frac{\pi}{2}\right) \cup \left(\frac{3 \pi}{2}, 2 \pi\right]$$
x in Union(Interval.Ropen(0, pi/2), Interval.Lopen(3*pi/2, 2*pi))