Mister Exam
tan(3*x+pi/6)<0 inequation
The solution
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/ pi\ tan|3*x + --| < 0 \ 6 /
$$\tan{\left(3 x + \frac{\pi}{6} \right)} < 0$$
tan(3*x + pi/6) < 0
Detail solution
Given the inequality:
$$\tan{\left(3 x + \frac{\pi}{6} \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(3 x + \frac{\pi}{6} \right)} = 0$$
Solve:
Given the equation
$$\tan{\left(3 x + \frac{\pi}{6} \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\tan{\left(3 x + \frac{\pi}{6} \right)} = 0$$
This equation is transformed to
$$3 x + \frac{\pi}{6} = \pi n + \operatorname{atan}{\left(0 \right)}$$
Or
$$3 x + \frac{\pi}{6} = \pi n$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation
with the opposite sign, in total:
$$3 x = \pi n - \frac{\pi}{6}$$
Divide both parts of the equation by
$$3$$
$$x_{1} = \frac{\pi n}{3} - \frac{\pi}{18}$$
$$x_{1} = \frac{\pi n}{3} - \frac{\pi}{18}$$
This roots
$$x_{1} = \frac{\pi n}{3} - \frac{\pi}{18}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{\pi n}{3} - \frac{\pi}{18}\right) + - \frac{1}{10}$$
=
$$\frac{\pi n}{3} - \frac{\pi}{18} - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(3 x + \frac{\pi}{6} \right)} < 0$$
$$\tan{\left(3 \left(\frac{\pi n}{3} - \frac{\pi}{18} - \frac{1}{10}\right) + \frac{\pi}{6} \right)} < 0$$
the solution of our inequality is:
$$x < \frac{\pi n}{3} - \frac{\pi}{18}$$
$$\tan{\left(3 x + \frac{\pi}{6} \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(3 x + \frac{\pi}{6} \right)} = 0$$
Solve:
Given the equation
$$\tan{\left(3 x + \frac{\pi}{6} \right)} = 0$$
- this is the simplest trigonometric equation
with the change of sign in 0
We get:
$$\tan{\left(3 x + \frac{\pi}{6} \right)} = 0$$
This equation is transformed to
$$3 x + \frac{\pi}{6} = \pi n + \operatorname{atan}{\left(0 \right)}$$
Or
$$3 x + \frac{\pi}{6} = \pi n$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation
with the opposite sign, in total:
$$3 x = \pi n - \frac{\pi}{6}$$
Divide both parts of the equation by
$$3$$
$$x_{1} = \frac{\pi n}{3} - \frac{\pi}{18}$$
$$x_{1} = \frac{\pi n}{3} - \frac{\pi}{18}$$
This roots
$$x_{1} = \frac{\pi n}{3} - \frac{\pi}{18}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{\pi n}{3} - \frac{\pi}{18}\right) + - \frac{1}{10}$$
=
$$\frac{\pi n}{3} - \frac{\pi}{18} - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(3 x + \frac{\pi}{6} \right)} < 0$$
$$\tan{\left(3 \left(\frac{\pi n}{3} - \frac{\pi}{18} - \frac{1}{10}\right) + \frac{\pi}{6} \right)} < 0$$
tan(-3/10 + pi*n) < 0
the solution of our inequality is:
$$x < \frac{\pi n}{3} - \frac{\pi}{18}$$
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