Solve for x the equation 5sin^2x+5sinxcosx=3 .

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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We'll re-write the right side such as 3 = 3*1 = 3*[(sin x)^2 + (cos x)^2]

We'll re-write the equation:

5(sin x)^2 + 5sin x*cos x - 3(sin x)^2 - 3(cos x)^2 = 0

We'll combine like terms:

2(sin x)^2 + 5sin x*cos x- 3(cos x)^2 = 0

We'll divide the entire equation by (cos x)^2;

2(sin x)^2/(cos x)^2 + 5sin x*cos x/(cos x)^2 - 3 = 0

We'll replace (sin x)^2/(cos x)^2 by (tan x)^2

2(tan x)^2 + 5tan x - 3 = 0

We'll replace tan x by t:

2t^2 + 5t - 3 = 0

We'll apply quadratic formula:

t1 = [-5+sqrt(25 - 24)]/4

t1 = (-5+1)/4

t1 = -1

t2 = -3/2

tan x = t1 => tan x = -1 => x1 = pi - arctan 1 + k*pi

x1 = pi - pi/4 + k*pi

x1 = 3pi/4 + k*pi

x2 = pi - arctan (3/2) + k*pi

The solutions of the equation are: {3pi/4 + k*pi}U{pi - arctan (3/2) + k*pi}.

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