Mister Exam
sin(x-3)>0 inequation
The solution
Detail solution
Given the inequality:
$$\sin{\left(x - 3 \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x - 3 \right)} = 0$$
Solve:
Given the equation
$$\sin{\left(x - 3 \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\sin{\left(x - 3 \right)} = 0$$
This equation is transformed to
$$x - 3 = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x - 3 = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$x - 3 = 2 \pi n$$
$$x - 3 = 2 \pi n + \pi$$
, where n - is a integer
Move
$$-3$$
to right part of the equation
with the opposite sign, in total:
$$x = 2 \pi n + 3$$
$$x = 2 \pi n + 3 + \pi$$
$$x_{1} = 2 \pi n + 3$$
$$x_{2} = 2 \pi n + 3 + \pi$$
$$x_{1} = 2 \pi n + 3$$
$$x_{2} = 2 \pi n + 3 + \pi$$
This roots
$$x_{1} = 2 \pi n + 3$$
$$x_{2} = 2 \pi n + 3 + \pi$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n + 3\right) + - \frac{1}{10}$$
=
$$2 \pi n + \frac{29}{10}$$
substitute to the expression
$$\sin{\left(x - 3 \right)} > 0$$
$$\sin{\left(\left(2 \pi n + \frac{29}{10}\right) - 3 \right)} > 0$$
Then
$$x < 2 \pi n + 3$$
no execute
one of the solutions of our inequality is:
$$x > 2 \pi n + 3 \wedge x < 2 \pi n + 3 + \pi$$
$$\sin{\left(x - 3 \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x - 3 \right)} = 0$$
Solve:
Given the equation
$$\sin{\left(x - 3 \right)} = 0$$
- this is the simplest trigonometric equation
with the change of sign in 0
We get:
$$\sin{\left(x - 3 \right)} = 0$$
This equation is transformed to
$$x - 3 = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x - 3 = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$x - 3 = 2 \pi n$$
$$x - 3 = 2 \pi n + \pi$$
, where n - is a integer
Move
$$-3$$
to right part of the equation
with the opposite sign, in total:
$$x = 2 \pi n + 3$$
$$x = 2 \pi n + 3 + \pi$$
$$x_{1} = 2 \pi n + 3$$
$$x_{2} = 2 \pi n + 3 + \pi$$
$$x_{1} = 2 \pi n + 3$$
$$x_{2} = 2 \pi n + 3 + \pi$$
This roots
$$x_{1} = 2 \pi n + 3$$
$$x_{2} = 2 \pi n + 3 + \pi$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n + 3\right) + - \frac{1}{10}$$
=
$$2 \pi n + \frac{29}{10}$$
substitute to the expression
$$\sin{\left(x - 3 \right)} > 0$$
$$\sin{\left(\left(2 \pi n + \frac{29}{10}\right) - 3 \right)} > 0$$
sin(-1/10 + 2*pi*n) > 0
Then
$$x < 2 \pi n + 3$$
no execute
one of the solutions of our inequality is:
$$x > 2 \pi n + 3 \wedge x < 2 \pi n + 3 + \pi$$
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x1 x2
Solving inequality on a graph
Rapid solution
[src]
/ / / ___________________\\ / / ___________________\\ \ | | / /sin(3)\\ | / 2 2 || | / /sin(3)\\ | / 2 2 || | And|x < -I*|I*|2*pi + atan|------|| + log\\/ cos (3) + sin (3) /|, -I*|I*|pi + atan|------|| + log\\/ cos (3) + sin (3) /| < x| \ \ \ \cos(3)// / \ \ \cos(3)// / /
$$x < - i \left(\log{\left(\sqrt{\sin^{2}{\left(3 \right)} + \cos^{2}{\left(3 \right)}} \right)} + i \left(\operatorname{atan}{\left(\frac{\sin{\left(3 \right)}}{\cos{\left(3 \right)}} \right)} + 2 \pi\right)\right) \wedge - i \left(\log{\left(\sqrt{\sin^{2}{\left(3 \right)} + \cos^{2}{\left(3 \right)}} \right)} + i \left(\operatorname{atan}{\left(\frac{\sin{\left(3 \right)}}{\cos{\left(3 \right)}} \right)} + \pi\right)\right) < x$$
(-i*(i*(pi + atan(sin(3)/cos(3))) + log(sqrt(cos(3)^2 + sin(3)^2))) < x)∧(x < -i*(i*(2*pi + atan(sin(3)/cos(3))) + log(sqrt(cos(3)^2 + sin(3)^2))))