Mister Exam
tan(x+pi/6)>1/(sqrt(3)) inequation
The solution
You have entered
[src]
/ pi\ 1
tan|x + --| > -----
\ 6 / ___
\/ 3
$$\tan{\left(x + \frac{\pi}{6} \right)} > \frac{1}{\sqrt{3}}$$
tan(x + pi/6) > 1/(sqrt(3))
Detail solution
Given the inequality:
$$\tan{\left(x + \frac{\pi}{6} \right)} > \frac{1}{\sqrt{3}}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(x + \frac{\pi}{6} \right)} = \frac{1}{\sqrt{3}}$$
Solve:
Given the equation
$$\tan{\left(x + \frac{\pi}{6} \right)} = \frac{1}{\sqrt{3}}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x + \frac{\pi}{6} = \pi n + \operatorname{atan}{\left(\frac{\sqrt{3}}{3} \right)}$$
Or
$$x + \frac{\pi}{6} = \pi n + \frac{\pi}{6}$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation
with the opposite sign, in total:
$$x = \pi n$$
$$x_{1} = \pi n$$
$$x_{1} = \pi n$$
This roots
$$x_{1} = \pi n$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\pi n + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(x + \frac{\pi}{6} \right)} > \frac{1}{\sqrt{3}}$$
$$\tan{\left(\left(\pi n - \frac{1}{10}\right) + \frac{\pi}{6} \right)} > \frac{1}{\sqrt{3}}$$
Then
$$x < \pi n$$
no execute
the solution of our inequality is:
$$x > \pi n$$
$$\tan{\left(x + \frac{\pi}{6} \right)} > \frac{1}{\sqrt{3}}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(x + \frac{\pi}{6} \right)} = \frac{1}{\sqrt{3}}$$
Solve:
Given the equation
$$\tan{\left(x + \frac{\pi}{6} \right)} = \frac{1}{\sqrt{3}}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x + \frac{\pi}{6} = \pi n + \operatorname{atan}{\left(\frac{\sqrt{3}}{3} \right)}$$
Or
$$x + \frac{\pi}{6} = \pi n + \frac{\pi}{6}$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation
with the opposite sign, in total:
$$x = \pi n$$
$$x_{1} = \pi n$$
$$x_{1} = \pi n$$
This roots
$$x_{1} = \pi n$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\pi n + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(x + \frac{\pi}{6} \right)} > \frac{1}{\sqrt{3}}$$
$$\tan{\left(\left(\pi n - \frac{1}{10}\right) + \frac{\pi}{6} \right)} > \frac{1}{\sqrt{3}}$$
___
/ 1 pi \ \/ 3
tan|- -- + -- + pi*n| > -----
\ 10 6 / 3
Then
$$x < \pi n$$
no execute
the solution of our inequality is:
$$x > \pi n$$
_____
/
-------ο-------
x1
Solving inequality on a graph