Mister Exam

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cos(t)<1/2 inequation

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cos(t) < 1/2

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

cos(t) < 1/2

Detail solution

Given the inequality:

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

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

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

Solve:
Given the equation

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

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

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

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

t = \pi n + \frac{\pi}{3}

t = \pi n - \frac{2 \pi}{3}

, where n - is a integer

t_{1} = \pi n + \frac{\pi}{3}

t_{2} = \pi n - \frac{2 \pi}{3}

t_{1} = \pi n + \frac{\pi}{3}

t_{2} = \pi n - \frac{2 \pi}{3}

This roots

t_{1} = \pi n + \frac{\pi}{3}

t_{2} = \pi n - \frac{2 \pi}{3}

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

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

substitute to the expression

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

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

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

one of the solutions of our inequality is:

t < \pi n + \frac{\pi}{3}

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

Other solutions will get with the changeover to the next point
etc.
The answer:

t < \pi n + \frac{\pi}{3}

t > \pi n - \frac{2 \pi}{3}

Solving inequality on a graph

Rapid solution [src]

   /pi          5*pi\
And|-- < t, t < ----|
   \3            3  /

\frac{\pi}{3} < t \wedge t < \frac{5 \pi}{3}

(pi/3 < t)∧(t < 5*pi/3)

Rapid solution 2 [src]

 pi  5*pi 
(--, ----)
 3    3

t\ in\ \left(\frac{\pi}{3}, \frac{5 \pi}{3}\right)

t in Interval.open(pi/3, 5*pi/3)