Mister Exam

Lang:

cos(x)<(-sqrt(3))/2 inequation

You have entered [src]

            ___ 
         -\/ 3  
cos(x) < -------
            2

\cos{\left(x \right)} < \frac{\left(-1\right) \sqrt{3}}{2}

cos(x) < (-sqrt(3))/2

Detail solution

Given the inequality:

\cos{\left(x \right)} < \frac{\left(-1\right) \sqrt{3}}{2}

To solve this inequality, we must first solve the corresponding equation:

\cos{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{2}

Solve:
Given the equation

\cos{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{2}

- this is the simplest trigonometric equation
This equation is transformed to

x = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} \right)}

x = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} \right)}

x = \pi n + \frac{5 \pi}{6}

x = \pi n - \frac{\pi}{6}

, where n - is a integer

x_{1} = \pi n + \frac{5 \pi}{6}

x_{2} = \pi n - \frac{\pi}{6}

x_{1} = \pi n + \frac{5 \pi}{6}

x_{2} = \pi n - \frac{\pi}{6}

This roots

x_{1} = \pi n + \frac{5 \pi}{6}

x_{2} = \pi n - \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}

\left(\pi n + \frac{5 \pi}{6}\right) + - \frac{1}{10}

\pi n - \frac{1}{10} + \frac{5 \pi}{6}

substitute to the expression

\cos{\left(x \right)} < \frac{\left(-1\right) \sqrt{3}}{2}

\cos{\left(\pi n - \frac{1}{10} + \frac{5 \pi}{6} \right)} < \frac{\left(-1\right) \sqrt{3}}{2}

                            ___ 
    /  1    pi       \   -\/ 3  
-sin|- -- + -- + pi*n| < -------
    \  10   3        /      2

but

                            ___ 
    /  1    pi       \   -\/ 3  
-sin|- -- + -- + pi*n| > -------
    \  10   3        /      2

Then

x < \pi n + \frac{5 \pi}{6}

no execute
one of the solutions of our inequality is:

x > \pi n + \frac{5 \pi}{6} \wedge x < \pi n - \frac{\pi}{6}

         _____  
        /     \  
-------ο-------ο-------
       x1      x2

Solving inequality on a graph

Rapid solution [src]

   /5*pi          7*pi\
And|---- < x, x < ----|
   \ 6             6  /

\frac{5 \pi}{6} < x \wedge x < \frac{7 \pi}{6}

(5*pi/6 < x)∧(x < 7*pi/6)

Rapid solution 2 [src]

 5*pi  7*pi 
(----, ----)
  6     6

x\ in\ \left(\frac{5 \pi}{6}, \frac{7 \pi}{6}\right)

x in Interval.open(5*pi/6, 7*pi/6)