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sin(t)<-sqrt(2)/2 inequation

You have entered [src]

            ___ 
         -\/ 2  
sin(t) < -------
            2

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

sin(t) < (-sqrt(2))/2

Detail solution

Given the inequality:

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

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

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

Solve:
Given the equation

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

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

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

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

t = 2 \pi n - \frac{\pi}{4}

t = 2 \pi n + \frac{5 \pi}{4}

, where n - is a integer

t_{1} = 2 \pi n - \frac{\pi}{4}

t_{2} = 2 \pi n + \frac{5 \pi}{4}

t_{1} = 2 \pi n - \frac{\pi}{4}

t_{2} = 2 \pi n + \frac{5 \pi}{4}

This roots

t_{1} = 2 \pi n - \frac{\pi}{4}

t_{2} = 2 \pi n + \frac{5 \pi}{4}

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(2 \pi n - \frac{\pi}{4}\right) + - \frac{1}{10}

2 \pi n - \frac{\pi}{4} - \frac{1}{10}

substitute to the expression

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

\sin{\left(2 \pi n - \frac{\pi}{4} - \frac{1}{10} \right)} < \frac{\left(-1\right) \sqrt{2}}{2}

                            ___ 
    /1    pi         \   -\/ 2  
-sin|-- + -- - 2*pi*n| < -------
    \10   4          /      2

one of the solutions of our inequality is:

t < 2 \pi n - \frac{\pi}{4}

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

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

t < 2 \pi n - \frac{\pi}{4}

t > 2 \pi n + \frac{5 \pi}{4}

Solving inequality on a graph

Rapid solution [src]

   /5*pi          7*pi\
And|---- < t, t < ----|
   \ 4             4  /

\frac{5 \pi}{4} < t \wedge t < \frac{7 \pi}{4}

(5*pi/4 < t)∧(t < 7*pi/4)

Rapid solution 2 [src]

 5*pi  7*pi 
(----, ----)
  4     4

t\ in\ \left(\frac{5 \pi}{4}, \frac{7 \pi}{4}\right)

t in Interval.open(5*pi/4, 7*pi/4)