kow to solve the inequality trigonometry for 2 sin x cos x - sin x + 2 cos x -1

a. First, transform the trig inequality into a
product:

F = (sin x + 1)(2cos x -1) <
0

b. Solve  u = (sin x +1) = 0 and  v = (2 cos x - 1) =
0

sin x = 1 --> x = 3Pi/2;  cos x = 1/2 --> x
= Pi/3 and x = 4Pi/3

c. Set up a Sign Table: Variation of u
and v within period 2Pi

x          |  0         
Pi/3        Pi        3Pi/2        4Pi/3           
2Pi

u          |        +            +         +    0     
+               +

v          |        +     0      -
         -            -      0        +

F          |       
No          yes       yes        yes             No

The
sign of F is the sign of the product u*v

The solution set
of the inequality is the interval ( Pi/3, 4Pi/3)

Note 1: To
solve a trig inequality, transform it into a product of basic trig inequalities. Then,
set up a Sign Table that shows all the solutions sets of these basic inequalities. The
combined solution set can be easily seen as the sign of a
product

Note 2. Don't try to discuss the signs of the 2
binomials since it will lead to confusion and error.