We'll re-write the given equation and we'll multiply by
cos x, having all terms to one side:
2(sin x)^2 + sin x -
2sin x*cos x - cos x = 0
We'll factorize the first 2 terms
by sin x and the last 2 terms by - cos x:
sin x(2 sin x +
1) - cos x(2 sin x + 1) = 0
We'll factorize by (2 sin x +
1):
(2 sin x + 1)(sin x - cos x) =
0
We'll put the first factor as
zero:
2 sin x + 1 = 0
We'll
subtract 1;
2sinx = -1
sin x =
-1/2
x = arcsin (-1/2)
The
sine function is negative in the 3rd and 4th quadrants:
x =
pi + pi/6
x = 7pi/6 (3rd
qudrant)
x = 2pi -
pi/6
x = 11pi/6 (4th
qudrant)
We'll put the next factor as
zero:
sin x - cos x = 0
This
is an homogeneous equation and we'll divide it by cos
x:
tan x - 1 = 0
tan x =
1
The function tangent is positive in the 1st and the 3rd
qudrants:
x = arctan
1
x = pi/4 (1st
quadrant)
x = pi+
pi/4
x = 5pi/4 (3rd
qudrant)
The complete set
of solutions of the equation, in the interval[0 , 2pi], are: {pi/4 ; 5pi/4 ; 7pi/6 ;
11pi/6}.
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