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Solving inequalities/ sint<√3/2

sint<√3/2 inequation

A inequation with variable

The solution

You have entered [src] 
           ___
         \/ 3 
sin(t) < -----
           2
sin⁡(t)<32\sin{\left(t \right)} < \frac{\sqrt{3}}{2} 
sin(t) < sqrt(3)/2
Detail solution
Given the inequality:
sin⁡(t)<32\sin{\left(t \right)} < \frac{\sqrt{3}}{2}
To solve this inequality, we must first solve the corresponding equation:
sin⁡(t)=32\sin{\left(t \right)} = \frac{\sqrt{3}}{2}
Solve:
Given the equation
sin⁡(t)=32\sin{\left(t \right)} = \frac{\sqrt{3}}{2}
- this is the simplest trigonometric equation
This equation is transformed to
t=2πn+asin⁡(32)t = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{3}}{2} \right)}
t=2πn−asin⁡(32)+πt = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{3}}{2} \right)} + \pi
Or
t=2πn+π3t = 2 \pi n + \frac{\pi}{3}
t=2πn+2π3t = 2 \pi n + \frac{2 \pi}{3}
, where n - is a integer
t1=2πn+π3t_{1} = 2 \pi n + \frac{\pi}{3}
t2=2πn+2π3t_{2} = 2 \pi n + \frac{2 \pi}{3}
t1=2πn+π3t_{1} = 2 \pi n + \frac{\pi}{3}
t2=2πn+2π3t_{2} = 2 \pi n + \frac{2 \pi}{3}
This roots
t1=2πn+π3t_{1} = 2 \pi n + \frac{\pi}{3}
t2=2πn+2π3t_{2} = 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:
t0<t1t_{0} < t_{1}
For example, let's take the point
t0=t1−110t_{0} = t_{1} - \frac{1}{10}
=
(2πn+π3)+−110\left(2 \pi n + \frac{\pi}{3}\right) + - \frac{1}{10}
=
2πn−110+π32 \pi n - \frac{1}{10} + \frac{\pi}{3}
substitute to the expression
sin⁡(t)<32\sin{\left(t \right)} < \frac{\sqrt{3}}{2}
sin⁡(2πn−110+π3)<32\sin{\left(2 \pi n - \frac{1}{10} + \frac{\pi}{3} \right)} < \frac{\sqrt{3}}{2}
                            ___
   /  1    pi         \   \/ 3 
sin|- -- + -- + 2*pi*n| < -----
   \  10   3          /     2

one of the solutions of our inequality is:
t<2πn+π3t < 2 \pi n + \frac{\pi}{3}
 _____           _____          
      \         /
-------ο-------ο-------
       t1      t2

Other solutions will get with the changeover to the next point
etc.
The answer:
t<2πn+π3t < 2 \pi n + \frac{\pi}{3}
t>2πn+2π3t > 2 \pi n + \frac{2 \pi}{3}
Solving inequality on a graph
0-60-50-40-30-20-101020304050602-2
Rapid solution 2 [src] 
    pi     2*pi       
[0, --) U (----, 2*pi]
    3       3
t in [0,π3)∪(2π3,2π]t\ in\ \left[0, \frac{\pi}{3}\right) \cup \left(\frac{2 \pi}{3}, 2 \pi\right] 
t in Union(Interval.Ropen(0, pi/3), Interval.Lopen(2*pi/3, 2*pi))
Rapid solution [src] 
  /   /            pi\     /           2*pi    \\
Or|And|0 <= t, t < --|, And|t <= 2*pi, ---- < t||
  \   \            3 /     \            3      //
(0≤t∧t<π3)∨(t≤2π∧2π3<t)\left(0 \leq t \wedge t < \frac{\pi}{3}\right) \vee \left(t \leq 2 \pi \wedge \frac{2 \pi}{3} < t\right) 
((0 <= t)∧(t < pi/3))∨((t <= 2*pi)∧(2*pi/3 < t))
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