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1/(cos(2x))+tan(2x)=3cos(2x)

Solución

\frac{1}{cos ( 2 x )} + tan (2 x) = 3 cos (2 x)

x = \frac{0.72972 \dots}{2} + πn, x = \frac{π}{2} - \frac{0.72972 \dots}{2} + πn

Pasos de solución

\frac{1}{cos ( 2 x )} + tan (2 x) = 3 cos (2 x)

Restar

3 cos (2 x)

de ambos lados

\frac{1}{cos ( 2 x )} + tan (2 x) - 3 cos (2 x) = 0

Simplificar

\frac{1}{cos ( 2 x )} + tan (2 x) - 3 cos (2 x) : \frac{1 + tan ( 2 x ) cos ( 2 x ) - 3 cos ^{2} ( 2 x )}{cos ( 2 x )}

\frac{1 + tan ( 2 x ) cos ( 2 x ) - 3 cos ^{2} ( 2 x )}{cos ( 2 x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

1 + tan (2 x) cos (2 x) - 3 cos^{2} (2 x) = 0

Expresar con seno, coseno

1 + \frac{sin ( 2 x )}{cos ( 2 x )} cos (2 x) - 3 cos^{2} (2 x) = 0

Simplificar

1 + \frac{sin ( 2 x )}{cos ( 2 x )} cos (2 x) - 3 cos^{2} (2 x) : 1 + sin (2 x) - 3 cos^{2} (2 x)

1 + sin (2 x) - 3 cos^{2} (2 x) = 0

Sumar

3 cos^{2} (2 x)

a ambos lados

1 + sin (2 x) = 3 cos^{2} (2 x)

Elevar al cuadrado ambos lados

(1 + sin (2 x))^{2} = (3 cos^{2} (2 x))^{2}

Restar

(3 cos^{2} (2 x))^{2}

de ambos lados

(1 + sin (2 x))^{2} - 9 cos^{4} (2 x) = 0

Factorizar

(1 + sin (2 x))^{2} - 9 cos^{4} (2 x) : (1 + sin (2 x) + 3 cos^{2} (2 x)) (1 + sin (2 x) - 3 cos^{2} (2 x))

(1 + sin (2 x) + 3 cos^{2} (2 x)) (1 + sin (2 x) - 3 cos^{2} (2 x)) = 0

Resolver cada parte por separado

1 + sin (2 x) + 3 cos^{2} (2 x) = 0 or 1 + sin (2 x) - 3 cos^{2} (2 x) = 0

1 + sin (2 x) + 3 cos^{2} (2 x) = 0 : x = \frac{3 π}{4} + πn

1 + sin (2 x) - 3 cos^{2} (2 x) = 0 : x = \frac{arcsin ( \frac{2}{3} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{2}{3} )}{2} + πn, x = \frac{3 π}{4} + πn

Combinar toda las soluciones

x = \frac{3 π}{4} + πn, x = \frac{arcsin ( \frac{2}{3} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{2}{3} )}{2} + πn

Verificar las soluciones sustituyendo en la ecuación original

x = \frac{arcsin ( \frac{2}{3} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{2}{3} )}{2} + πn

Mostrar soluciones en forma decimal

x = \frac{0.72972 \dots}{2} + πn, x = \frac{π}{2} - \frac{0.72972 \dots}{2} + πn

tan (θ) = - 24

tan^{2} (θ) + 5 tan (θ) + 2 = 0

cos (z) = - 2

sin (2 x) = sin (x), x, \in [0, 2 π]

cos (2 x) + cos (- x) = 0