Mister Exam
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How to use it?
- Inequation:
-
sin(x)tg2x>0
- logx(x3–1)≤logx(x3+2x–4)
- 4(x−1,5)−1,2≥6x-1
- log3(9x)*log4(64x)/(5x^2-|x|)<=0
-
Identical expressions
- sin(x)tg2x> zero
- sinus of (x)tg2x greater than 0
- sinus of (x)tg2x greater than zero
- sinxtg2x>0
-
Similar expressions
- sin(x)(tg(2x))+1>0
- sinxtg2x>0
sin(x)tg2x>0 inequation
The solution
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[src]
sin(x)*tan(2*x) > 0
$$\sin{\left(x \right)} \tan{\left(2 x \right)} > 0$$
sin(x)*tan(2*x) > 0
Detail solution
Given the inequality:
$$\sin{\left(x \right)} \tan{\left(2 x \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x \right)} \tan{\left(2 x \right)} = 0$$
Solve:
Given the equation:
$$\sin{\left(x \right)} \tan{\left(2 x \right)} = 0$$
Transform
$$\sin{\left(x \right)} \tan{\left(2 x \right)} = 0$$
$$\sin{\left(x \right)} \tan{\left(2 x \right)} = 0$$
Consider each factor separately
$$\sin{\left(x \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\sin{\left(x \right)} = 0$$
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$x = 2 \pi n$$
$$x = 2 \pi n + \pi$$
, where n - is a integer
$$\tan{\left(2 x \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\tan{\left(2 x \right)} = 0$$
This equation is transformed to
$$2 x = \pi n + \operatorname{atan}{\left(0 \right)}$$
Or
$$2 x = \pi n$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
get the intermediate answer:
$$x = 2 \pi n$$
$$x = 2 \pi n + \pi$$
$$x = \frac{\pi n}{2}$$
The final answer:
$$x_{1} = 2 \pi n$$
$$x_{2} = 2 \pi n + \pi$$
$$x_{3} = \frac{\pi n}{2}$$
$$x_{1} = 2 \pi n$$
$$x_{2} = 2 \pi n + \pi$$
$$x_{3} = \frac{\pi n}{2}$$
$$x_{1} = 2 \pi n$$
$$x_{2} = 2 \pi n + \pi$$
$$x_{3} = \frac{\pi n}{2}$$
This roots
$$x_{1} = 2 \pi n$$
$$x_{2} = 2 \pi n + \pi$$
$$x_{3} = \frac{\pi n}{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(x \right)} \tan{\left(2 x \right)} > 0$$
$$\sin{\left(2 \pi n - \frac{1}{10} \right)} \tan{\left(2 \cdot \left(2 \pi n - \frac{1}{10}\right) \right)} > 0$$
one of the solutions of our inequality is:
$$x < 2 \pi n$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 2 \pi n$$
$$x > 2 \pi n + \pi \wedge x < \frac{\pi n}{2}$$
$$\sin{\left(x \right)} \tan{\left(2 x \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x \right)} \tan{\left(2 x \right)} = 0$$
Solve:
Given the equation:
$$\sin{\left(x \right)} \tan{\left(2 x \right)} = 0$$
Transform
$$\sin{\left(x \right)} \tan{\left(2 x \right)} = 0$$
$$\sin{\left(x \right)} \tan{\left(2 x \right)} = 0$$
Consider each factor separately
Step
$$\sin{\left(x \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\sin{\left(x \right)} = 0$$
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$x = 2 \pi n$$
$$x = 2 \pi n + \pi$$
, where n - is a integer
Step
$$\tan{\left(2 x \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\tan{\left(2 x \right)} = 0$$
This equation is transformed to
$$2 x = \pi n + \operatorname{atan}{\left(0 \right)}$$
Or
$$2 x = \pi n$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
get the intermediate answer:
$$x = 2 \pi n$$
$$x = 2 \pi n + \pi$$
$$x = \frac{\pi n}{2}$$
The final answer:
$$x_{1} = 2 \pi n$$
$$x_{2} = 2 \pi n + \pi$$
$$x_{3} = \frac{\pi n}{2}$$
$$x_{1} = 2 \pi n$$
$$x_{2} = 2 \pi n + \pi$$
$$x_{3} = \frac{\pi n}{2}$$
$$x_{1} = 2 \pi n$$
$$x_{2} = 2 \pi n + \pi$$
$$x_{3} = \frac{\pi n}{2}$$
This roots
$$x_{1} = 2 \pi n$$
$$x_{2} = 2 \pi n + \pi$$
$$x_{3} = \frac{\pi n}{2}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(x \right)} \tan{\left(2 x \right)} > 0$$
$$\sin{\left(2 \pi n - \frac{1}{10} \right)} \tan{\left(2 \cdot \left(2 \pi n - \frac{1}{10}\right) \right)} > 0$$
sin(1/10)*tan(1/5) > 0
one of the solutions of our inequality is:
$$x < 2 \pi n$$
_____ _____
\ / \
-------ο-------ο-------ο-------
x_1 x_2 x_3Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < 2 \pi n$$
$$x > 2 \pi n + \pi \wedge x < \frac{\pi n}{2}$$
Solving inequality on a graph
Rapid solution 2
[src]
pi pi 3*pi 5*pi 3*pi 7*pi
(0, --) U (--, ----) U (----, ----) U (----, 2*pi)
4 2 4 4 2 4
$$x\ in\ \left(0, \frac{\pi}{4}\right) \cup \left(\frac{\pi}{2}, \frac{3 \pi}{4}\right) \cup \left(\frac{5 \pi}{4}, \frac{3 \pi}{2}\right) \cup \left(\frac{7 \pi}{4}, 2 \pi\right)$$
x in Union(Interval.open(0, pi/4), Interval.open(pi/2, 3*pi/4), Interval.open(5*pi/4, 3*pi/2), Interval.open(7*pi/4, 2*pi))
Rapid solution
[src]
/ / pi\ /pi 3*pi\ /5*pi 3*pi\ /7*pi \\ Or|And|0 < x, x < --|, And|-- < x, x < ----|, And|---- < x, x < ----|, And|---- < x, x < 2*pi|| \ \ 4 / \2 4 / \ 4 2 / \ 4 //
$$\left(0 < x \wedge x < \frac{\pi}{4}\right) \vee \left(\frac{\pi}{2} < x \wedge x < \frac{3 \pi}{4}\right) \vee \left(\frac{5 \pi}{4} < x \wedge x < \frac{3 \pi}{2}\right) \vee \left(\frac{7 \pi}{4} < x \wedge x < 2 \pi\right)$$
((0 < x)∧(x < pi/4))∨((pi/2 < x)∧(x < 3*pi/4))∨((5*pi/4 < x)∧(x < 3*pi/2))∨((7*pi/4 < x)∧(x < 2*pi))
The graph