Mister Exam

Lang:

-2*cos(x/3)>1 inequation

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      /x\    
-2*cos|-| > 1
      \3/

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

-2*cos(x/3) > 1

Detail solution

Given the inequality:

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

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

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

Solve:
Given the equation

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

- this is the simplest trigonometric equation

Divide both parts of the equation by -2

The equation is transformed to

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

This equation is transformed to

\frac{x}{3} = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}

\frac{x}{3} = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}

\frac{x}{3} = \pi n + \frac{2 \pi}{3}

\frac{x}{3} = \pi n - \frac{\pi}{3}

, where n - is a integer
Divide both parts of the equation by

\frac{1}{3}

x_{1} = 3 \pi n + 2 \pi

x_{2} = 3 \pi n - \pi

x_{1} = 3 \pi n + 2 \pi

x_{2} = 3 \pi n - \pi

This roots

x_{1} = 3 \pi n + 2 \pi

x_{2} = 3 \pi n - \pi

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(3 \pi n + 2 \pi\right) + - \frac{1}{10}

3 \pi n - \frac{1}{10} + 2 \pi

substitute to the expression

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

- 2 \cos{\left(\frac{3 \pi n - \frac{1}{10} + 2 \pi}{3} \right)} > 1

     /  1    pi       \    
2*sin|- -- + -- + pi*n| > 1
     \  30   6        /

Then

x < 3 \pi n + 2 \pi

no execute
one of the solutions of our inequality is:

x > 3 \pi n + 2 \pi \wedge x < 3 \pi n - \pi

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

Solving inequality on a graph

Rapid solution [src]

And(2*pi < x, x < 4*pi)

2 \pi < x \wedge x < 4 \pi

(2*pi < x)∧(x < 4*pi)

Rapid solution 2 [src]

(2*pi, 4*pi)

x\ in\ \left(2 \pi, 4 \pi\right)

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