Mister Exam
tan(x/7-5pi/6)<-√3 inequation
The solution
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/x 5*pi\ ___ tan|- - ----| < -\/ 3 \7 6 /
$$\tan{\left(\frac{x}{7} - \frac{5 \pi}{6} \right)} < - \sqrt{3}$$
tan(x/7 - 5*pi/6) < -sqrt(3)
Detail solution
Given the inequality:
$$\tan{\left(\frac{x}{7} - \frac{5 \pi}{6} \right)} < - \sqrt{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(\frac{x}{7} - \frac{5 \pi}{6} \right)} = - \sqrt{3}$$
Solve:
Given the equation
$$\tan{\left(\frac{x}{7} - \frac{5 \pi}{6} \right)} = - \sqrt{3}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{7} + \frac{\pi}{6} = \pi n + \operatorname{atan}{\left(- \sqrt{3} \right)}$$
Or
$$\frac{x}{7} + \frac{\pi}{6} = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{7} = \pi n - \frac{\pi}{2}$$
Divide both parts of the equation by
$$\frac{1}{7}$$
$$x_{1} = 7 \pi n - \frac{7 \pi}{2}$$
$$x_{1} = 7 \pi n - \frac{7 \pi}{2}$$
This roots
$$x_{1} = 7 \pi n - \frac{7 \pi}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(7 \pi n - \frac{7 \pi}{2}\right) + - \frac{1}{10}$$
=
$$7 \pi n - \frac{7 \pi}{2} - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(\frac{x}{7} - \frac{5 \pi}{6} \right)} < - \sqrt{3}$$
$$\tan{\left(\frac{7 \pi n - \frac{7 \pi}{2} - \frac{1}{10}}{7} - \frac{5 \pi}{6} \right)} < - \sqrt{3}$$
the solution of our inequality is:
$$x < 7 \pi n - \frac{7 \pi}{2}$$
$$\tan{\left(\frac{x}{7} - \frac{5 \pi}{6} \right)} < - \sqrt{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(\frac{x}{7} - \frac{5 \pi}{6} \right)} = - \sqrt{3}$$
Solve:
Given the equation
$$\tan{\left(\frac{x}{7} - \frac{5 \pi}{6} \right)} = - \sqrt{3}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{7} + \frac{\pi}{6} = \pi n + \operatorname{atan}{\left(- \sqrt{3} \right)}$$
Or
$$\frac{x}{7} + \frac{\pi}{6} = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{7} = \pi n - \frac{\pi}{2}$$
Divide both parts of the equation by
$$\frac{1}{7}$$
$$x_{1} = 7 \pi n - \frac{7 \pi}{2}$$
$$x_{1} = 7 \pi n - \frac{7 \pi}{2}$$
This roots
$$x_{1} = 7 \pi n - \frac{7 \pi}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(7 \pi n - \frac{7 \pi}{2}\right) + - \frac{1}{10}$$
=
$$7 \pi n - \frac{7 \pi}{2} - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(\frac{x}{7} - \frac{5 \pi}{6} \right)} < - \sqrt{3}$$
$$\tan{\left(\frac{7 \pi n - \frac{7 \pi}{2} - \frac{1}{10}}{7} - \frac{5 \pi}{6} \right)} < - \sqrt{3}$$
/1 pi \ ___
-tan|-- + -- - pi*n| < -\/ 3
\70 3 / the solution of our inequality is:
$$x < 7 \pi n - \frac{7 \pi}{2}$$
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