Mister Exam
cos(x)tg(2x)<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
$$\cos{\left(x \right)} \tan{\left(2 x \right)} < 0$$
$$\cos{\left(- \frac{\pi}{2} - \frac{1}{10} \right)} \tan{\left(2 \left(- \frac{\pi}{2} - \frac{1}{10}\right) \right)} < 0$$
but
Then
$$x < - \frac{\pi}{2}$$
no execute
one of the solutions of our inequality is:
$$x > - \frac{\pi}{2} \wedge x < 0$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x > - \frac{\pi}{2} \wedge x < 0$$
$$x > \frac{\pi}{2}$$
$$\cos{\left(x \right)} \tan{\left(2 x \right)} < 0$$
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
but
sin(1/10)*tan(1/5) > 0
Then
$$x < - \frac{\pi}{2}$$
no execute
one of the solutions of our inequality is:
$$x > - \frac{\pi}{2} \wedge x < 0$$
_____ _____
/ \ /
-------ο-------ο-------ο-------
x2 x1 x3Other solutions will get with the changeover to the next point
etc.
The answer:
$$x > - \frac{\pi}{2} \wedge x < 0$$
$$x > \frac{\pi}{2}$$
Solving inequality on a graph
Rapid solution
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/ /pi pi\ /pi 3*pi\ / 5*pi\ /7*pi \\ Or|And|-- < x, x < --|, And|-- < x, x < ----|, And|pi < x, x < ----|, And|---- < x, x < 2*pi|| \ \4 2 / \2 4 / \ 4 / \ 4 //
$$\left(\frac{\pi}{4} < x \wedge x < \frac{\pi}{2}\right) \vee \left(\frac{\pi}{2} < x \wedge x < \frac{3 \pi}{4}\right) \vee \left(\pi < x \wedge x < \frac{5 \pi}{4}\right) \vee \left(\frac{7 \pi}{4} < x \wedge x < 2 \pi\right)$$
((pi < x)∧(x < 5*pi/4))∨((pi/4 < x)∧(x < pi/2))∨((pi/2 < x)∧(x < 3*pi/4))∨((7*pi/4 < x)∧(x < 2*pi))
Rapid solution 2
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pi pi pi 3*pi 5*pi 7*pi (--, --) U (--, ----) U (pi, ----) U (----, 2*pi) 4 2 2 4 4 4
$$x\ in\ \left(\frac{\pi}{4}, \frac{\pi}{2}\right) \cup \left(\frac{\pi}{2}, \frac{3 \pi}{4}\right) \cup \left(\pi, \frac{5 \pi}{4}\right) \cup \left(\frac{7 \pi}{4}, 2 \pi\right)$$
x in Union(Interval.open(pi/4, pi/2), Interval.open(pi/2, 3*pi/4), Interval.open(pi, 5*pi/4), Interval.open(7*pi/4, 2*pi))