Mister Exam

ctg<√3 inequation

A inequation with variable

The solution

You have entered [src]
           ___
cot(x) < \/ 3 
$$\cot{\left(x \right)} < \sqrt{3}$$
cot(x) < sqrt(3)
Detail solution
Given the inequality:
$$\cot{\left(x \right)} < \sqrt{3}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cot{\left(x \right)} = \sqrt{3}$$
Solve:
Given the equation
$$\cot{\left(x \right)} = \sqrt{3}$$
transform
$$\cot{\left(x \right)} - \sqrt{3} - 1 = 0$$
$$\cot{\left(x \right)} - \sqrt{3} - 1 = 0$$
Do replacement
$$w = \cot{\left(x \right)}$$
Expand brackets in the left part
-1 + w - sqrt3 = 0

Move free summands (without w)
from left part to right part, we given:
$$w - \sqrt{3} = 1$$
Divide both parts of the equation by (w - sqrt(3))/w
w = 1 / ((w - sqrt(3))/w)

We get the answer: w = 1 + sqrt(3)
do backward replacement
$$\cot{\left(x \right)} = w$$
substitute w:
$$x_{1} = \frac{\pi}{6}$$
$$x_{1} = \frac{\pi}{6}$$
This roots
$$x_{1} = \frac{\pi}{6}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{\pi}{6}$$
=
$$- \frac{1}{10} + \frac{\pi}{6}$$
substitute to the expression
$$\cot{\left(x \right)} < \sqrt{3}$$
$$\cot{\left(- \frac{1}{10} + \frac{\pi}{6} \right)} < \sqrt{3}$$
   /1    pi\     ___
tan|-- + --| < \/ 3 
   \10   3 /   

but
   /1    pi\     ___
tan|-- + --| > \/ 3 
   \10   3 /   

Then
$$x < \frac{\pi}{6}$$
no execute
the solution of our inequality is:
$$x > \frac{\pi}{6}$$
         _____  
        /
-------ο-------
       x1
Rapid solution [src]
   /pi            \
And|-- < x, x < pi|
   \6             /
$$\frac{\pi}{6} < x \wedge x < \pi$$
(x < pi)∧(pi/6 < x)
Rapid solution 2 [src]
 pi     
(--, pi)
 6      
$$x\ in\ \left(\frac{\pi}{6}, \pi\right)$$
x in Interval.open(pi/6, pi)