Monday, November 12, 2012

Given that 2sinx+1=2sin^2x/cosx+sinx/cosx what is x.

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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