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4cos^2(x)+3sin^2(x)-4=-tan^2(x)

Solución

4 cos^{2} (x) + 3 sin^{2} (x) - 4 = - tan^{2} (x)

Solución

x = 2 πn, x = π + 2 πn

Grados

x = 0^{\circ} + 36 0^{\circ} n, x = 18 0^{\circ} + 36 0^{\circ} n

Pasos de solución

4 cos^{2} (x) + 3 sin^{2} (x) - 4 = - tan^{2} (x)

Restar

- tan^{2} (x)

de ambos lados

4 cos^{2} (x) + 3 sin^{2} (x) - 4 + tan^{2} (x) = 0

Re-escribir usando identidades trigonométricas

- 4 + tan^{2} (x) + 3 sin^{2} (x) + 4 cos^{2} (x)

Utilizar la identidad pitagórica:

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

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

= - 4 + tan^{2} (x) + 3 sin^{2} (x) + 4 (1 - sin^{2} (x))

Simplificar

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

- 4 + tan^{2} (x) + 3 sin^{2} (x) + 4 (1 - sin^{2} (x))

Expandir

4 (1 - sin^{2} (x)) : 4 - 4 sin^{2} (x)

4 (1 - sin^{2} (x))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 4, b = 1, c = sin^{2} (x)

= 4 \cdot 1 - 4 sin^{2} (x)

Multiplicar los numeros:

4 \cdot 1 = 4

= 4 - 4 sin^{2} (x)

= - 4 + tan^{2} (x) + 3 sin^{2} (x) + 4 - 4 sin^{2} (x)

Simplificar

- 4 + tan^{2} (x) + 3 sin^{2} (x) + 4 - 4 sin^{2} (x) : tan^{2} (x) - sin^{2} (x)

- 4 + tan^{2} (x) + 3 sin^{2} (x) + 4 - 4 sin^{2} (x)

Agrupar términos semejantes

= tan^{2} (x) + 3 sin^{2} (x) - 4 sin^{2} (x) - 4 + 4

Sumar elementos similares:

3 sin^{2} (x) - 4 sin^{2} (x) = - sin^{2} (x)

= tan^{2} (x) - sin^{2} (x) - 4 + 4

- 4 + 4 = 0

= tan^{2} (x) - sin^{2} (x)

= tan^{2} (x) - sin^{2} (x)

= tan^{2} (x) - sin^{2} (x)

- sin^{2} (x) + tan^{2} (x) = 0

Factorizar

- sin^{2} (x) + tan^{2} (x) : (tan (x) + sin (x)) (tan (x) - sin (x))

- sin^{2} (x) + tan^{2} (x)

Aplicar la siguiente regla para binomios al cuadrado:

x^{2} - y^{2} = (x + y) (x - y)

tan^{2} (x) - sin^{2} (x) = (tan (x) + sin (x)) (tan (x) - sin (x))

= (tan (x) + sin (x)) (tan (x) - sin (x))

(tan (x) + sin (x)) (tan (x) - sin (x)) = 0

Resolver cada parte por separado

tan (x) + sin (x) = 0 or tan (x) - sin (x) = 0

tan (x) + sin (x) = 0 : x = 2 πn, x = π + 2 πn

tan (x) + sin (x) = 0

Expresar con seno, coseno

sin (x) + tan (x)

Utilizar la identidad trigonométrica básica:

tan (x) = \frac{sin ( x )}{cos ( x )}

= sin (x) + \frac{sin ( x )}{cos ( x )}

Simplificar

sin (x) + \frac{sin ( x )}{cos ( x )} : \frac{sin ( x ) cos ( x ) + sin ( x )}{cos ( x )}

sin (x) + \frac{sin ( x )}{cos ( x )}

Convertir a fracción:

sin (x) = \frac{sin ( x ) cos ( x )}{cos ( x )}

= \frac{sin ( x ) cos ( x )}{cos ( x )} + \frac{sin ( x )}{cos ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{sin ( x ) cos ( x ) + sin ( x )}{cos ( x )}

= \frac{sin ( x ) cos ( x ) + sin ( x )}{cos ( x )}

\frac{sin ( x ) + cos ( x ) sin ( x )}{cos ( x )} = 0

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

sin (x) + cos (x) sin (x) = 0

Factorizar

sin (x) + cos (x) sin (x) : sin (x) (cos (x) + 1)

sin (x) + cos (x) sin (x)

Factorizar el termino común

sin (x)

= sin (x) (1 + cos (x))

sin (x) (cos (x) + 1) = 0

Resolver cada parte por separado

sin (x) = 0 or cos (x) + 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

Soluciones generales para

sin (x) = 0

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

cos (x) + 1 = 0 : x = π + 2 πn

cos (x) + 1 = 0

Mover

1

al lado derecho

cos (x) + 1 = 0

Restar

1

de ambos lados

cos (x) + 1 - 1 = 0 - 1

Simplificar

cos (x) = - 1

cos (x) = - 1

Soluciones generales para

cos (x) = - 1

cos (x)

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = π + 2 πn

x = π + 2 πn

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn

tan (x) - sin (x) = 0 : x = 2 πn, x = π + 2 πn

tan (x) - sin (x) = 0

Expresar con seno, coseno

- sin (x) + tan (x)

Utilizar la identidad trigonométrica básica:

tan (x) = \frac{sin ( x )}{cos ( x )}

= - sin (x) + \frac{sin ( x )}{cos ( x )}

Simplificar

- sin (x) + \frac{sin ( x )}{cos ( x )} : \frac{- sin ( x ) cos ( x ) + sin ( x )}{cos ( x )}

- sin (x) + \frac{sin ( x )}{cos ( x )}

Convertir a fracción:

sin (x) = \frac{sin ( x ) cos ( x )}{cos ( x )}

= - \frac{sin ( x ) cos ( x )}{cos ( x )} + \frac{sin ( x )}{cos ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{- sin ( x ) cos ( x ) + sin ( x )}{cos ( x )}

= \frac{- sin ( x ) cos ( x ) + sin ( x )}{cos ( x )}

\frac{sin ( x ) - cos ( x ) sin ( x )}{cos ( x )} = 0

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

sin (x) - cos (x) sin (x) = 0

Factorizar

sin (x) - cos (x) sin (x) : - sin (x) (cos (x) - 1)

sin (x) - cos (x) sin (x)

Factorizar el termino común

- sin (x)

= - sin (x) (- 1 + cos (x))

- sin (x) (cos (x) - 1) = 0

Resolver cada parte por separado

sin (x) = 0 or cos (x) - 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) = 0

Soluciones generales para

sin (x) = 0

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = 0 + 2 πn, x = π + 2 πn

x = 0 + 2 πn, x = π + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn, x = π + 2 πn

cos (x) - 1 = 0 : x = 2 πn

cos (x) - 1 = 0

Mover

1

al lado derecho

cos (x) - 1 = 0

Sumar

1

a ambos lados

cos (x) - 1 + 1 = 0 + 1

Simplificar

cos (x) = 1

cos (x) = 1

Soluciones generales para

cos (x) = 1

cos (x)

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

x = 0 + 2 πn

x = 0 + 2 πn

Resolver

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn

Ejemplos populares

tan^2(x)-2=3tan(x)

tan^{2} (x) - 2 = 3 tan (x)

sqrt(3)cos(x)+sin(x)=sqrt(3)

3 cos (x) + sin (x) = 3

2tan^2(θ)-2=0

2 tan^{2} (θ) - 2 = 0

sin^2(x)=2-sin(x)

sin^{2} (x) = 2 - sin (x)

2csc(x)cos(x)=csc(x)

2 csc (x) cos (x) = csc (x)