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Popular >

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

Solución

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

Restar

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

de ambos lados

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

Expresar con seno, coseno

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

Utilizar la identidad trigonométrica básica:

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

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

Simplificar

- sin^{2} (x) - (\frac{sin ( x )}{cos ( x )})^{2} + sin^{2} (x) (\frac{sin ( x )}{cos ( x )})^{2} : \frac{- sin ^{2} ( x ) cos ^{2} ( x ) - sin ^{2} ( x ) + sin ^{4} ( x )}{cos ^{2} ( x )}

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

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

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

Aplicar las leyes de los exponentes:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

sin^{2} (x) (\frac{sin ( x )}{cos ( x )})^{2} = \frac{sin ^{4} ( x )}{cos ^{2} ( x )}

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

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

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

Aplicar las leyes de los exponentes:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

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

Multiplicar fracciones:

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

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

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

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

Aplicar las leyes de los exponentes:

a^{b} \cdot a^{c} = a^{b + c}

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

= sin^{2 + 2} (x)

Sumar:

2 + 2 = 4

= sin^{4} (x)

= \frac{sin ^{4} ( x )}{cos ^{2} ( x )}

= - sin^{2} (x) - \frac{sin ^{2} ( x )}{cos ^{2} ( x )} + \frac{sin ^{4} ( x )}{cos ^{2} ( x )}

Combinar las fracciones usando el mínimo común denominador:

\frac{- sin ^{2} ( x ) + sin ^{4} ( x )}{cos ^{2} ( x )}

Aplicar la regla

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

= \frac{- sin ^{2} ( x ) + sin ^{4} ( x )}{cos ^{2} ( x )}

= - sin^{2} (x) + \frac{sin ^{4} ( x ) - sin ^{2} ( x )}{cos ^{2} ( x )}

Convertir a fracción:

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

= - \frac{sin ^{2} ( x ) cos ^{2} ( x )}{cos ^{2} ( x )} + \frac{- sin ^{2} ( x ) + sin ^{4} ( x )}{cos ^{2} ( x )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{- sin ^{2} ( x ) cos ^{2} ( x ) - sin ^{2} ( x ) + sin ^{4} ( x )}{cos ^{2} ( x )}

= \frac{- sin ^{2} ( x ) cos ^{2} ( x ) - sin ^{2} ( x ) + sin ^{4} ( x )}{cos ^{2} ( x )}

\frac{- sin ^{2} ( x ) + sin ^{4} ( x ) - cos ^{2} ( x ) sin ^{2} ( x )}{cos ^{2} ( x )} = 0

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

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

Factorizar

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

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

Aplicar las leyes de los exponentes:

a^{b + c} = a^{b} a^{c}

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

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

Factorizar el termino común

- sin^{2} (x)

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

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

Resolver cada parte por separado

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

sin^{2} (x) = 0 : x = 2 πn, x = π + 2 πn

sin^{2} (x) = 0

Aplicar la regla

x^{n} = 0 \Rightarrow x = 0

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

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

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica del ángulo doble:

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

= 1 + cos (2 x)

1 + cos (2 x) = 0

Mover

1

al lado derecho

1 + cos (2 x) = 0

Restar

1

de ambos lados

1 + cos (2 x) - 1 = 0 - 1

Simplificar

cos (2 x) = - 1

cos (2 x) = - 1

Soluciones generales para

cos (2 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}

2 x = π + 2 πn

2 x = π + 2 πn

Resolver

2 x = π + 2 πn : x = \frac{π}{2} + πn

2 x = π + 2 πn

Dividir ambos lados entre

2

2 x = π + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{π}{2} + \frac{2 πn}{2}

Simplificar

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

Combinar toda las soluciones

x = 2 πn, x = π + 2 πn, x = \frac{π}{2} + πn

Siendo que la ecuación esta indefinida para:

\frac{π}{2} + πn

x = 2 πn, x = π + 2 πn

Ejemplos populares

solvefor θ,cot(θ)=-pi/6 resolver para

θ, cot (θ) = - \frac{π}{6}

tan(x)+cos(x)=0

tan (x) + cos (x) = 0

tan(θ)=sin(2θ)

tan (θ) = sin (2 θ)

sin(θ_{1})=-10/13

sin (θ_{1}) = - \frac{10}{13}

sin(a^{-1})=(2.6)/(0.1733)

sin (a^{- 1}) = \frac{2.6}{0.1733}