Mister Exam

Lang:

cost<-sqrt(3)/2 inequation

You have entered [src]

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

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

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

Detail solution

Given the inequality:

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

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

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

Solve:
Given the equation

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

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

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

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

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

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

, where n - is a integer

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

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

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

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

This roots

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

t_{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:

t_{0} < t_{1}

For example, let's take the point

t_{0} = t_{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(t \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

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

no execute
one of the solutions of our inequality is:

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

         _____  
        /     \  
-------ο-------ο-------
       t1      t2

Solving inequality on a graph

Rapid solution [src]

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

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

(5*pi/6 < t)∧(t < 7*pi/6)

Rapid solution 2 [src]

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

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

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