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+ inequation

Solve the inequality!

A inequation with variable

The solution

You have entered [src]

tan(x) < cos(pi)

$$\tan{\left(x \right)} < \cos{\left(\pi \right)}$$

tan(x) < cos(pi)

Detail solution

Given the inequality:
$$\tan{\left(x \right)} < \cos{\left(\pi \right)}$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(x \right)} = \cos{\left(\pi \right)}$$
Solve:
Given the equation
$$\tan{\left(x \right)} = \cos{\left(\pi \right)}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(-1 \right)}$$
Or
$$x = \pi n - \frac{\pi}{4}$$
, where n - is a integer
$$x_{1} = \pi n - \frac{\pi}{4}$$
$$x_{1} = \pi n - \frac{\pi}{4}$$
This roots
$$x_{1} = \pi n - \frac{\pi}{4}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n - \frac{\pi}{4}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{\pi}{4} - \frac{1}{10}$$
substitute to the expression
$$\tan{\left(x \right)} < \cos{\left(\pi \right)}$$
$$\tan{\left(\pi n - \frac{\pi}{4} - \frac{1}{10} \right)} < \cos{\left(\pi \right)}$$

    /1    pi       \     
-tan|-- + -- - pi*n| < -1
    \10   4        /     

the solution of our inequality is:
$$x < \pi n - \frac{\pi}{4}$$

 _____          
      \    
-------ο-------
       x1

Solving inequality on a graph

Rapid solution [src]

   /pi          3*pi\
And|-- < x, x < ----|
   \2            4  /

$$\frac{\pi}{2} < x \wedge x < \frac{3 \pi}{4}$$

(pi/2 < x)∧(x < 3*pi/4)

Rapid solution 2 [src]

 pi  3*pi 
(--, ----)
 2    4   

$$x\ in\ \left(\frac{\pi}{2}, \frac{3 \pi}{4}\right)$$

x in Interval.open(pi/2, 3*pi/4)