Mister Exam
cosxtg2x>0 inequation
The solution
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cos(x)*tan(2*x) > 0
$$\cos{\left(x \right)} \tan{\left(2 x \right)} > 0$$
cos(x)*tan(2*x) > 0
Detail solution
Given the inequality:
$$\cos{\left(x \right)} \tan{\left(2 x \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(x \right)} \tan{\left(2 x \right)} = 0$$
Solve:
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = \frac{\pi}{2}$$
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = \frac{\pi}{2}$$
This roots
$$x_{2} = - \frac{\pi}{2}$$
$$x_{1} = 0$$
$$x_{3} = \frac{\pi}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{\pi}{2} - \frac{1}{10}$$
=
$$- \frac{\pi}{2} - \frac{1}{10}$$
substitute to the expression
$$\cos{\left(x \right)} \tan{\left(2 x \right)} > 0$$
$$\cos{\left(- \frac{\pi}{2} - \frac{1}{10} \right)} \tan{\left(2 \left(- \frac{\pi}{2} - \frac{1}{10}\right) \right)} > 0$$
one of the solutions of our inequality is:
$$x < - \frac{\pi}{2}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < - \frac{\pi}{2}$$
$$x > 0 \wedge x < \frac{\pi}{2}$$
$$\cos{\left(x \right)} \tan{\left(2 x \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(x \right)} \tan{\left(2 x \right)} = 0$$
Solve:
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = \frac{\pi}{2}$$
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = \frac{\pi}{2}$$
This roots
$$x_{2} = - \frac{\pi}{2}$$
$$x_{1} = 0$$
$$x_{3} = \frac{\pi}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{\pi}{2} - \frac{1}{10}$$
=
$$- \frac{\pi}{2} - \frac{1}{10}$$
substitute to the expression
$$\cos{\left(x \right)} \tan{\left(2 x \right)} > 0$$
$$\cos{\left(- \frac{\pi}{2} - \frac{1}{10} \right)} \tan{\left(2 \left(- \frac{\pi}{2} - \frac{1}{10}\right) \right)} > 0$$
sin(1/10)*tan(1/5) > 0
one of the solutions of our inequality is:
$$x < - \frac{\pi}{2}$$
_____ _____
\ / \
-------ο-------ο-------ο-------
x2 x1 x3Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < - \frac{\pi}{2}$$
$$x > 0 \wedge x < \frac{\pi}{2}$$
Solving inequality on a graph
Rapid solution
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/ / pi\ /5*pi 3*pi\ /3*pi 7*pi\ /3*pi \\ Or|And|0 < x, x < --|, And|---- < x, x < ----|, And|---- < x, x < ----|, And|---- < x, x < pi|| \ \ 4 / \ 4 2 / \ 2 4 / \ 4 //
$$\left(0 < x \wedge x < \frac{\pi}{4}\right) \vee \left(\frac{5 \pi}{4} < x \wedge x < \frac{3 \pi}{2}\right) \vee \left(\frac{3 \pi}{2} < x \wedge x < \frac{7 \pi}{4}\right) \vee \left(\frac{3 \pi}{4} < x \wedge x < \pi\right)$$
((0 < x)∧(x < pi/4))∨((x < pi)∧(3*pi/4 < x))∨((5*pi/4 < x)∧(x < 3*pi/2))∨((3*pi/2 < x)∧(x < 7*pi/4))
Rapid solution 2
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pi 3*pi 5*pi 3*pi 3*pi 7*pi
(0, --) U (----, pi) U (----, ----) U (----, ----)
4 4 4 2 2 4
$$x\ in\ \left(0, \frac{\pi}{4}\right) \cup \left(\frac{3 \pi}{4}, \pi\right) \cup \left(\frac{5 \pi}{4}, \frac{3 \pi}{2}\right) \cup \left(\frac{3 \pi}{2}, \frac{7 \pi}{4}\right)$$
x in Union(Interval.open(0, pi/4), Interval.open(3*pi/4, pi), Interval.open(5*pi/4, 3*pi/2), Interval.open(3*pi/2, 7*pi/4))