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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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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