SolutionsWhat is x if 5sin^2x + 3sinxcosx + cos^2x = 3 ?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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To determine x means to find the angle x from the given identity. We'll transform the given identity into a homogenous equation by substituting 3 by 3*1 = 3[(sin x)^2 + (cos x)^2] and moving all terms to one side.

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

We'll combine like terms:

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

Since cos x is different from zero, we'll divide the entire equation by (cos x)^2:

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

According to the rule, the ratio sin x/cos x = tan x.

2(tan x)^2 + 3tan x - 2 = 0

We'll substitute tan x = t:

2t^2 + 3t - 2 = 0

We'll apply the quadratic formula:

t1 = [-3+sqrt(9+16)]/2

t1 = (-3+5)/2

t1 = 1

t2 = (-5-3)/2

t2 = -4

We'll put tan x = t1:

tan x = 1

x = arctan 1 + k*pi

x = pi/4 + k*pi

tan x = t2

tan x = -4

x = - arctan (4) + k*pi

The solutions of the equation are the values of x angle:

{pi/4 + k*pi} U {- arctan (4) + k*pi}.

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