Find t: 4cos^2t-1 = sin^2t+sint*cost.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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To solve for t means to find the angle t from the given identity. We'll transform the given identity into a homogenous equation by substituting 1 by (sin t)^2 + (cos t)^2 = 1 and moving all terms to one side.

(sin t)^2 + sint*cost -  4(cos t)^2 + (sin t)^2 + (cos t)^2 = 0

We'll combine like terms:

2(sin t)^2 + sint*cost - 3(cos t)^2 = 0

Since cos t is different from zero, we'll divide the entire equation by (cos t)^2:

2(sin t)^2/(cos t)^2 + sint*cost/(cos t)^2 - 3 = 0

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

2 (tan t)^2 + tan t - 3 = 0

We'll substitute tan t = x:

2x^2 + x - 3 = 0

We'll apply the quadratic formula:

x1 = [-1+sqrt(1+24)]/4

x1 = (-1+5)/4

x1 = 1

x2 = (-1-5)/4

x2 = -3/2

We'll put tan t = x1:

tan t = 1

t = arctan 1 + k*pi

t = pi/4 + k*pi

tan t = x2

tan t = -3/2

t = - arctan (3/2) + k*pi

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

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

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