Mister Exam
sin((x+pi)/4)>0 inequation
The solution
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/x + pi\ sin|------| > 0 \ 4 /
$$\sin{\left(\frac{x + \pi}{4} \right)} > 0$$
sin((x + pi)/4) > 0
Detail solution
Given the inequality:
$$\sin{\left(\frac{x + \pi}{4} \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(\frac{x + \pi}{4} \right)} = 0$$
Solve:
Given the equation
$$\sin{\left(\frac{x + \pi}{4} \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\sin{\left(\frac{x + \pi}{4} \right)} = 0$$
This equation is transformed to
$$\frac{x}{4} + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$\frac{x}{4} + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$\frac{x}{4} + \frac{\pi}{4} = 2 \pi n$$
$$\frac{x}{4} + \frac{\pi}{4} = 2 \pi n + \pi$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{4} = 2 \pi n - \frac{\pi}{4}$$
$$\frac{x}{4} = 2 \pi n + \frac{3 \pi}{4}$$
Divide both parts of the equation by
$$\frac{1}{4}$$
$$x_{1} = 8 \pi n - \pi$$
$$x_{2} = 8 \pi n + 3 \pi$$
$$x_{1} = 8 \pi n - \pi$$
$$x_{2} = 8 \pi n + 3 \pi$$
This roots
$$x_{1} = 8 \pi n - \pi$$
$$x_{2} = 8 \pi n + 3 \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(8 \pi n - \pi\right) + - \frac{1}{10}$$
=
$$8 \pi n - \pi - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(\frac{x + \pi}{4} \right)} > 0$$
$$\sin{\left(\frac{\left(8 \pi n - \pi - \frac{1}{10}\right) + \pi}{4} \right)} > 0$$
Then
$$x < 8 \pi n - \pi$$
no execute
one of the solutions of our inequality is:
$$x > 8 \pi n - \pi \wedge x < 8 \pi n + 3 \pi$$
$$\sin{\left(\frac{x + \pi}{4} \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(\frac{x + \pi}{4} \right)} = 0$$
Solve:
Given the equation
$$\sin{\left(\frac{x + \pi}{4} \right)} = 0$$
- this is the simplest trigonometric equation
with the change of sign in 0
We get:
$$\sin{\left(\frac{x + \pi}{4} \right)} = 0$$
This equation is transformed to
$$\frac{x}{4} + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$\frac{x}{4} + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$\frac{x}{4} + \frac{\pi}{4} = 2 \pi n$$
$$\frac{x}{4} + \frac{\pi}{4} = 2 \pi n + \pi$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{4} = 2 \pi n - \frac{\pi}{4}$$
$$\frac{x}{4} = 2 \pi n + \frac{3 \pi}{4}$$
Divide both parts of the equation by
$$\frac{1}{4}$$
$$x_{1} = 8 \pi n - \pi$$
$$x_{2} = 8 \pi n + 3 \pi$$
$$x_{1} = 8 \pi n - \pi$$
$$x_{2} = 8 \pi n + 3 \pi$$
This roots
$$x_{1} = 8 \pi n - \pi$$
$$x_{2} = 8 \pi n + 3 \pi$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(8 \pi n - \pi\right) + - \frac{1}{10}$$
=
$$8 \pi n - \pi - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(\frac{x + \pi}{4} \right)} > 0$$
$$\sin{\left(\frac{\left(8 \pi n - \pi - \frac{1}{10}\right) + \pi}{4} \right)} > 0$$
sin(-1/40 + 2*pi*n) > 0
Then
$$x < 8 \pi n - \pi$$
no execute
one of the solutions of our inequality is:
$$x > 8 \pi n - \pi \wedge x < 8 \pi n + 3 \pi$$
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Solving inequality on a graph
Rapid solution
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Or(And(0 <= x, x < 3*pi), And(x <= 8*pi, 7*pi < x))
$$\left(0 \leq x \wedge x < 3 \pi\right) \vee \left(x \leq 8 \pi \wedge 7 \pi < x\right)$$
((0 <= x)∧(x < 3*pi))∨((x <= 8*pi)∧(7*pi < x))
Rapid solution 2
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[0, 3*pi) U (7*pi, 8*pi]
$$x\ in\ \left[0, 3 \pi\right) \cup \left(7 \pi, 8 \pi\right]$$
x in Union(Interval.Ropen(0, 3*pi), Interval.Lopen(7*pi, 8*pi))