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Popular >

sin^2(x)=2tan^2(x)

Solución

sin^{2} (x) = 2 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

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

Restar

2 tan^{2} (x)

de ambos lados

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

Factorizar

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

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

Reescribir

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

como

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

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

Aplicar las leyes de los exponentes:

a = (a)^{2}

2 = (2)^{2}

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

Aplicar las leyes de los exponentes:

a^{m} b^{m} = (ab)^{m}

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

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

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

Aplicar la siguiente regla para binomios al cuadrado:

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

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

= (sin (x) + 2 tan (x)) (sin (x) - 2 tan (x))

(sin (x) + 2 tan (x)) (sin (x) - 2 tan (x)) = 0

Resolver cada parte por separado

sin (x) + 2 tan (x) = 0 or sin (x) - 2 tan (x) = 0

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

sin (x) + 2 tan (x) = 0

Expresar con seno, coseno

sin (x) + 2 tan (x)

Utilizar la identidad trigonométrica básica:

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

= sin (x) + 2 \frac{sin ( x )}{cos ( x )}

Simplificar

sin (x) + 2 \frac{sin ( x )}{cos ( x )} : \frac{sin ( x ) cos ( x ) + 2 sin ( x )}{cos ( x )}

sin (x) + 2 \frac{sin ( x )}{cos ( x )}

Multiplicar

2 \frac{sin ( x )}{cos ( x )} : \frac{2 sin ( x )}{cos ( x )}

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

Multiplicar fracciones:

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

= \frac{sin ( x ) 2}{cos ( x )}

= sin (x) + \frac{2 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 ) 2}{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 ) 2}{cos ( x )}

= \frac{sin ( x ) cos ( x ) + 2 sin ( x )}{cos ( x )}

\frac{cos ( x ) sin ( x ) + sin ( x ) 2}{cos ( x )} = 0

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

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

Factorizar

cos (x) sin (x) + sin (x) 2 : sin (x) (cos (x) + 2)

cos (x) sin (x) + sin (x) 2

Factorizar el termino común

sin (x)

= sin (x) (cos (x) + 2)

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

Resolver cada parte por separado

sin (x) = 0 or cos (x) + 2 = 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) + 2 = 0 :

Sin solución

cos (x) + 2 = 0

Mover

2

al lado derecho

cos (x) + 2 = 0

Restar

2

de ambos lados

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

Simplificar

cos (x) = - 2

cos (x) = - 2

- 1 \leq cos (x) \leq 1

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn

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

sin (x) - 2 tan (x) = 0

Expresar con seno, coseno

sin (x) - 2 tan (x)

Utilizar la identidad trigonométrica básica:

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

= sin (x) - 2 \frac{sin ( x )}{cos ( x )}

Simplificar

sin (x) - 2 \frac{sin ( x )}{cos ( x )} : \frac{sin ( x ) cos ( x ) - 2 sin ( x )}{cos ( x )}

sin (x) - 2 \frac{sin ( x )}{cos ( x )}

Multiplicar

2 \frac{sin ( x )}{cos ( x )} : \frac{2 sin ( x )}{cos ( x )}

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

Multiplicar fracciones:

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

= \frac{sin ( x ) 2}{cos ( x )}

= sin (x) - \frac{2 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 ) 2}{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 ) 2}{cos ( x )}

= \frac{sin ( x ) cos ( x ) - 2 sin ( x )}{cos ( x )}

\frac{cos ( x ) sin ( x ) - sin ( x ) 2}{cos ( x )} = 0

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

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

Factorizar

cos (x) sin (x) - sin (x) 2 : sin (x) (cos (x) - 2)

cos (x) sin (x) - sin (x) 2

Factorizar el termino común

sin (x)

= sin (x) (cos (x) - 2)

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

Resolver cada parte por separado

sin (x) = 0 or cos (x) - 2 = 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) - 2 = 0 :

Sin solución

cos (x) - 2 = 0

Mover

2

al lado derecho

cos (x) - 2 = 0

Sumar

2

a ambos lados

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

Simplificar

cos (x) = 2

cos (x) = 2

- 1 \leq cos (x) \leq 1

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn

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(2cos(a)+1)(2cos(a)-1)=2cos^2(a)+1

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