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Popular Trigonometría >

sin^2(a)=((2tan(a)))/((1+tan^2(a)))

Solución

sin^{2} (a) = \frac{( 2 tan ( a ) )}{( 1 + tan ^{2} ( a ) )}

Solución

a = 2 πn, a = π + 2 πn, a = 1.10714 \dots + πn

Grados

a = 0^{\circ} + 36 0^{\circ} n, a = 18 0^{\circ} + 36 0^{\circ} n, a = 63.43494 \dots^{\circ} + 18 0^{\circ} n

Pasos de solución

sin^{2} (a) = \frac{( 2 tan ( a ) )}{( 1 + tan ^{2} ( a ) )}

Restar

\frac{2 tan ( a )}{1 + tan ^{2} ( a )}

de ambos lados

sin^{2} (a) - \frac{2 tan ( a )}{1 + tan ^{2} ( a )} = 0

Simplificar

sin^{2} (a) - \frac{2 tan ( a )}{1 + tan ^{2} ( a )} : \frac{sin ^{2} ( a ) ( 1 + tan ^{2} ( a ) ) - 2 tan ( a )}{1 + tan ^{2} ( a )}

sin^{2} (a) - \frac{2 tan ( a )}{1 + tan ^{2} ( a )}

Convertir a fracción:

sin^{2} (a) = \frac{sin ^{2} ( a ) ( 1 + tan ^{2} ( a ) )}{1 + tan ^{2} ( a )}

= \frac{sin ^{2} ( a ) ( 1 + tan ^{2} ( a ) )}{1 + tan ^{2} ( a )} - \frac{2 tan ( a )}{1 + tan ^{2} ( a )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{sin ^{2} ( a ) ( 1 + tan ^{2} ( a ) ) - 2 tan ( a )}{1 + tan ^{2} ( a )}

\frac{sin ^{2} ( a ) ( 1 + tan ^{2} ( a ) ) - 2 tan ( a )}{1 + tan ^{2} ( a )} = 0

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

sin^{2} (a) (1 + tan^{2} (a)) - 2 tan (a) = 0

Expresar con seno, coseno

(1 + tan^{2} (a)) sin^{2} (a) - 2 tan (a)

Utilizar la identidad trigonométrica básica:

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

= (1 + (\frac{sin ( a )}{cos ( a )})^{2}) sin^{2} (a) - 2 \cdot \frac{sin ( a )}{cos ( a )}

Simplificar

(1 + (\frac{sin ( a )}{cos ( a )})^{2}) sin^{2} (a) - 2 \cdot \frac{sin ( a )}{cos ( a )} : \frac{sin ^{2} ( a ) ( cos ^{2} ( a ) + sin ^{2} ( a ) ) - 2 sin ( a ) cos ( a )}{cos ^{2} ( a )}

(1 + (\frac{sin ( a )}{cos ( a )})^{2}) sin^{2} (a) - 2 \cdot \frac{sin ( a )}{cos ( a )}

(1 + (\frac{sin ( a )}{cos ( a )})^{2}) sin^{2} (a) = \frac{cos ^{2} ( a ) + sin ^{2} ( a )}{cos ^{2} ( a )} sin^{2} (a)

(1 + (\frac{sin ( a )}{cos ( a )})^{2}) sin^{2} (a)

Aplicar las leyes de los exponentes:

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

= sin^{2} (a) (\frac{sin ^{2} ( a )}{cos ^{2} ( a )} + 1)

Simplificar

1 + \frac{sin ^{2} ( a )}{cos ^{2} ( a )}

en una fracción:

\frac{cos ^{2} ( a ) + sin ^{2} ( a )}{cos ^{2} ( a )}

1 + \frac{sin ^{2} ( a )}{cos ^{2} ( a )}

Convertir a fracción:

1 = \frac{1 cos ^{2} ( a )}{cos ^{2} ( a )}

= \frac{1 \cdot cos ^{2} ( a )}{cos ^{2} ( a )} + \frac{sin ^{2} ( a )}{cos ^{2} ( a )}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{1 \cdot cos ^{2} ( a ) + sin ^{2} ( a )}{cos ^{2} ( a )}

Multiplicar:

1 \cdot cos^{2} (a) = cos^{2} (a)

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

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

2 \cdot \frac{sin ( a )}{cos ( a )} = \frac{2 sin ( a )}{cos ( a )}

2 \cdot \frac{sin ( a )}{cos ( a )}

Multiplicar fracciones:

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

= \frac{sin ( a ) \cdot 2}{cos ( a )}

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

Multiplicar

\frac{cos ^{2} ( a ) + sin ^{2} ( a )}{cos ^{2} ( a )} sin^{2} (a) : \frac{sin ^{2} ( a ) ( cos ^{2} ( a ) + sin ^{2} ( a ) )}{cos ^{2} ( a )}

\frac{cos ^{2} ( a ) + sin ^{2} ( a )}{cos ^{2} ( a )} sin^{2} (a)

Multiplicar fracciones:

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

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

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

Mínimo común múltiplo de

cos^{2} (a), cos (a) : cos^{2} (a)

cos^{2} (a), cos (a)

Mínimo común múltiplo (MCM)

Calcular una expresión que este compuesta de factores que aparezcan tanto en

cos^{2} (a)

cos (a)

= cos^{2} (a)

Reescribir las fracciones basandose en el mínimo común denominador

Multiplicar cada numerador por la misma cantidad necesaria para multiplicar el denominador correspondiente y convertirlo en el mínimo común denominador

Para

\frac{sin ( a ) \cdot 2}{cos ( a )} :

multiplicar el denominador y el numerador por

cos (a)

\frac{sin ( a ) \cdot 2}{cos ( a )} = \frac{sin ( a ) \cdot 2 cos ( a )}{cos ( a ) cos ( a )} = \frac{sin ( a ) \cdot 2 cos ( a )}{cos ^{2} ( a )}

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

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

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

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

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

Factorizar

(cos^{2} (a) + sin^{2} (a)) sin^{2} (a) - 2 cos (a) sin (a) : sin (a) (sin (a) (cos^{2} (a) + sin^{2} (a)) - 2 cos (a))

(cos^{2} (a) + sin^{2} (a)) sin^{2} (a) - 2 cos (a) sin (a)

Aplicar las leyes de los exponentes:

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

sin^{2} (a) = sin (a) sin (a)

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

Factorizar el termino común

sin (a)

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

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

Re-escribir usando identidades trigonométricas

sin (a) (sin (a) (cos^{2} (a) + sin^{2} (a)) - 2 cos (a))

Utilizar la identidad pitagórica:

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

= sin (a) (- 2 cos (a) + sin (a) \cdot 1)

Simplificar

sin (a) (- 2 cos (a) + sin (a) \cdot 1) : sin (a) (- 2 cos (a) + sin (a))

sin (a) (- 2 cos (a) + sin (a) \cdot 1)

Multiplicar:

sin (a) \cdot 1 = sin (a)

= sin (a) (sin (a) - 2 cos (a))

= sin (a) (- 2 cos (a) + sin (a))

sin (a) (- 2 cos (a) + sin (a)) = 0

Resolver cada parte por separado

sin (a) = 0 or - 2 cos (a) + sin (a) = 0

sin (a) = 0 : a = 2 πn, a = π + 2 πn

sin (a) = 0

Soluciones generales para

sin (a) = 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}

a = 0 + 2 πn, a = π + 2 πn

a = 0 + 2 πn, a = π + 2 πn

Resolver

a = 0 + 2 πn : a = 2 πn

a = 0 + 2 πn

0 + 2 πn = 2 πn

a = 2 πn

a = 2 πn, a = π + 2 πn

- 2 cos (a) + sin (a) = 0 : a = arctan (2) + πn

- 2 cos (a) + sin (a) = 0

Re-escribir usando identidades trigonométricas

- 2 cos (a) + sin (a) = 0

Dividir ambos lados entre

cos (a), cos (a) \neq = 0

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

Simplificar

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

Utilizar la identidad trigonométrica básica:

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

- 2 + tan (a) = 0

- 2 + tan (a) = 0

Desplace

2

a la derecha

- 2 + tan (a) = 0

Sumar

2

a ambos lados

- 2 + tan (a) + 2 = 0 + 2

Simplificar

tan (a) = 2

tan (a) = 2

Aplicar propiedades trigonométricas inversas

tan (a) = 2

Soluciones generales para

tan (a) = 2

tan (x) = a \Rightarrow x = arctan (a) + πn

a = arctan (2) + πn

a = arctan (2) + πn

Combinar toda las soluciones

a = 2 πn, a = π + 2 πn, a = arctan (2) + πn

Mostrar soluciones en forma decimal

a = 2 πn, a = π + 2 πn, a = 1.10714 \dots + πn

Gráfica

Ejemplos populares

solvefor a,sin^2(a)+cos^2(b)=1