What is the solution to the trigonometric inequality 2sin(x)+3> sin^2(x)

First of all, let's analyze this equation. There is no restriction for  

x  since the domain of a function  sin

( x )

is all real numbers.

Secondly, we can introduce an intermediary variable  y = sin ( x

and express this equation in terms of  y : y 2 2 y + 3  or y 2 − 2 y 3 = 0  

This equation has solutions: y 1 , 2 = 2 ± √ 4 + 12 2  or y 1 = 3  and  2 = − 1  

Now we have to take into consideration that  y = sin ( x )

and, as such, is restricted in values from  − 1  to  1

(inclusive).

Therefore, solution  y 1 = 3

would not lead to any value of  x .

The second solution  y 2 − 1

can be used to find  x : sin ( x ) = 1

implies that x= − π 2 + 2 π N , where  N  - any integer number or, in degrees, x = −

90 o +

360 o ⋅

N

This is a final solution to the problem