x=? sin^2x+sinx*cosx-4cos^2x+1=0

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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To find the angle x means to solve the given equation. We'll transform the given equation into a homogenous equation by substituting 1 by (sin x)^2 + (cos x)^2 = 1.

(sin x)^2 + sinx*cosx -  4(cos x)^2 + (sin x)^2 + (cos x)^2 = 0

We'll combine like terms:

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

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

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

We'll substitute tan x = t:

2t^2 + t - 3 = 0

We'll apply the quadratic formula:

t1 = [-1+sqrt(1+24)]/4

t1 = (-1+5)/4

t1 = 1

t2 = (-1-5)/4

t2 = -3/2

We'll put tan x = t1:

tan x = 1

x = arctan 1 + k*pi

x = pi/4 + k*pi

tan x = t2

tan x = -3/2

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

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

{pi/4 + k*pi} U {- arctan (3/2) + k*pi}

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