English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

tan(2x)=cos(2x),0<= x<= pi

Solución

tan (2 x) = cos (2 x), 0 \leq x \leq π

Solución

x = \frac{0.66623 \dots}{2}, x = \frac{π - 0.66623 \dots}{2}

Grados

x = 19.08635 \dots^{\circ}, x = 70.91364 \dots^{\circ}

Pasos de solución

tan (2 x) = cos (2 x), 0 \leq x \leq π

Restar

cos (2 x)

de ambos lados

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

Expresar con seno, coseno

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

Simplificar

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

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

Convertir a fracción:

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

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

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

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

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

cos (2 x) cos (2 x)

Aplicar las leyes de los exponentes:

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

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

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

Sumar:

1 + 1 = 2

= cos^{2} (2 x)

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

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

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

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

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

Sumar

cos^{2} (2 x)

a ambos lados

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

Elevar al cuadrado ambos lados

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

Restar

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

de ambos lados

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

Factorizar

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

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

Aplicar las leyes de los exponentes:

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

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

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

Aplicar la siguiente regla para binomios al cuadrado:

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

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

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

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

Resolver cada parte por separado

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

sin (2 x) + cos^{2} (2 x) = 0, 0 \leq x \leq π : x = \frac{π + arcsin ( \frac{5 - 1}{2} )}{2}, x = \frac{- arcsin ( \frac{5 - 1}{2} ) + 2 π}{2}

sin (2 x) + cos^{2} (2 x) = 0, 0 \leq x \leq π

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

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

Usando el método de sustitución

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

Sea:

sin (2 x) = u

1 + u - u^{2} = 0

1 + u - u^{2} = 0 : u = - \frac{- 1 + 5}{2}, u = \frac{1 + 5}{2}

1 + u - u^{2} = 0

Escribir en la forma binómica

a x^{2} + b x + c = 0

- u^{2} + u + 1 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

- u^{2} + u + 1 = 0

Formula general para ecuaciones de segundo grado:

Para

a = - 1, b = 1, c = 1

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 1 ) \cdot 1}{2 ( - 1 )}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 ( - 1 ) \cdot 1}{2 ( - 1 )}

1^{2} - 4 (- 1) \cdot 1 = 5

1^{2} - 4 (- 1) \cdot 1

Aplicar la regla

1^{a} = 1

1^{2} = 1

= 1 - 4 (- 1) \cdot 1

Aplicar la regla

- (- a) = a

= 1 + 4 \cdot 1 \cdot 1

Multiplicar los numeros:

4 \cdot 1 \cdot 1 = 4

= 1 + 4

Sumar:

1 + 4 = 5

= 5

u_{1, 2} = \frac{- 1 \pm 5}{2 ( - 1 )}

Separar las soluciones

u_{1} = \frac{- 1 + 5}{2 ( - 1 )}, u_{2} = \frac{- 1 - 5}{2 ( - 1 )}

u = \frac{- 1 + 5}{2 ( - 1 )} : - \frac{- 1 + 5}{2}

\frac{- 1 + 5}{2 ( - 1 )}

Quitar los parentesis:

(- a) = - a

= \frac{- 1 + 5}{- 2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{- 1 + 5}{- 2}

Aplicar las propiedades de las fracciones:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{- 1 + 5}{2}

u = \frac{- 1 - 5}{2 ( - 1 )} : \frac{1 + 5}{2}

\frac{- 1 - 5}{2 ( - 1 )}

Quitar los parentesis:

(- a) = - a

= \frac{- 1 - 5}{- 2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{- 1 - 5}{- 2}

Aplicar las propiedades de las fracciones:

\frac{- a}{- b} = \frac{a}{b}

- 1 - 5 = - (1 + 5)

= \frac{1 + 5}{2}

Las soluciones a la ecuación de segundo grado son:

u = - \frac{- 1 + 5}{2}, u = \frac{1 + 5}{2}

Sustituir en la ecuación

u = sin (2 x)

sin (2 x) = - \frac{- 1 + 5}{2}, sin (2 x) = \frac{1 + 5}{2}

sin (2 x) = - \frac{- 1 + 5}{2}, sin (2 x) = \frac{1 + 5}{2}

sin (2 x) = - \frac{- 1 + 5}{2}, 0 \leq x \leq π : x = \frac{π + arcsin ( \frac{5 - 1}{2} )}{2}, x = \frac{- arcsin ( \frac{5 - 1}{2} ) + 2 π}{2}

sin (2 x) = - \frac{- 1 + 5}{2}, 0 \leq x \leq π

Aplicar propiedades trigonométricas inversas

sin (2 x) = - \frac{- 1 + 5}{2}

Soluciones generales para

sin (2 x) = - \frac{- 1 + 5}{2}

sin (x) = - a \Rightarrow x = arcsin (- a) + 2 πn, x = π + arcsin (a) + 2 πn

2 x = arcsin (- \frac{- 1 + 5}{2}) + 2 πn, 2 x = π + arcsin (\frac{- 1 + 5}{2}) + 2 πn

2 x = arcsin (- \frac{- 1 + 5}{2}) + 2 πn, 2 x = π + arcsin (\frac{- 1 + 5}{2}) + 2 πn

Resolver

2 x = arcsin (- \frac{- 1 + 5}{2}) + 2 πn : x = - \frac{arcsin ( \frac{5 - 1}{2} )}{2} + πn

2 x = arcsin (- \frac{- 1 + 5}{2}) + 2 πn

Simplificar

arcsin (- \frac{- 1 + 5}{2}) + 2 πn : - arcsin (\frac{5 - 1}{2}) + 2 πn

arcsin (- \frac{- 1 + 5}{2}) + 2 πn

Utilizar la siguiente propiedad:

arcsin (- x) = - arcsin (x)

arcsin (- \frac{5 - 1}{2}) = - arcsin (\frac{5 - 1}{2})

= - arcsin (\frac{5 - 1}{2}) + 2 πn

2 x = - arcsin (\frac{5 - 1}{2}) + 2 πn

Dividir ambos lados entre

2

2 x = - arcsin (\frac{5 - 1}{2}) + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = - \frac{arcsin ( \frac{5 - 1}{2} )}{2} + \frac{2 πn}{2}

Simplificar

x = - \frac{arcsin ( \frac{5 - 1}{2} )}{2} + πn

x = - \frac{arcsin ( \frac{5 - 1}{2} )}{2} + πn

Resolver

2 x = π + arcsin (\frac{- 1 + 5}{2}) + 2 πn : x = \frac{π}{2} + \frac{arcsin ( \frac{- 1 + 5}{2} )}{2} + πn

2 x = π + arcsin (\frac{- 1 + 5}{2}) + 2 πn

Dividir ambos lados entre

2

2 x = π + arcsin (\frac{- 1 + 5}{2}) + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{π}{2} + \frac{arcsin ( \frac{- 1 + 5}{2} )}{2} + \frac{2 πn}{2}

Simplificar

x = \frac{π}{2} + \frac{arcsin ( \frac{- 1 + 5}{2} )}{2} + πn

x = \frac{π}{2} + \frac{arcsin ( \frac{- 1 + 5}{2} )}{2} + πn

x = - \frac{arcsin ( \frac{5 - 1}{2} )}{2} + πn, x = \frac{π}{2} + \frac{arcsin ( \frac{- 1 + 5}{2} )}{2} + πn

Soluciones para el rango

0 \leq x \leq π

x = \frac{π + arcsin ( \frac{5 - 1}{2} )}{2}, x = \frac{- arcsin ( \frac{5 - 1}{2} ) + 2 π}{2}

sin (2 x) = \frac{1 + 5}{2}, 0 \leq x \leq π :

Sin solución

sin (2 x) = \frac{1 + 5}{2}, 0 \leq x \leq π

- 1 \leq sin (x) \leq 1

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

x = \frac{π + arcsin ( \frac{5 - 1}{2} )}{2}, x = \frac{- arcsin ( \frac{5 - 1}{2} ) + 2 π}{2}

sin (2 x) - cos^{2} (2 x) = 0, 0 \leq x \leq π : x = \frac{arcsin ( \frac{5 - 1}{2} )}{2}, x = \frac{π - arcsin ( \frac{5 - 1}{2} )}{2}

sin (2 x) - cos^{2} (2 x) = 0, 0 \leq x \leq π

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

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

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

Poner los parentesis

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

Aplicar las reglas de los signos

- (- a) = a, - (a) = - a

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

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

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

Usando el método de sustitución

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

Sea:

sin (2 x) = u

- 1 + u + u^{2} = 0

- 1 + u + u^{2} = 0 : u = \frac{- 1 + 5}{2}, u = \frac{- 1 - 5}{2}

- 1 + u + u^{2} = 0

Escribir en la forma binómica

a x^{2} + b x + c = 0

u^{2} + u - 1 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

u^{2} + u - 1 = 0

Formula general para ecuaciones de segundo grado:

Para

a = 1, b = 1, c = - 1

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

1^{2} - 4 \cdot 1 \cdot (- 1) = 5

1^{2} - 4 \cdot 1 \cdot (- 1)

Aplicar la regla

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 1 \cdot (- 1)

Aplicar la regla

- (- a) = a

= 1 + 4 \cdot 1 \cdot 1

Multiplicar los numeros:

4 \cdot 1 \cdot 1 = 4

= 1 + 4

Sumar:

1 + 4 = 5

= 5

u_{1, 2} = \frac{- 1 \pm 5}{2 \cdot 1}

Separar las soluciones

u_{1} = \frac{- 1 + 5}{2 \cdot 1}, u_{2} = \frac{- 1 - 5}{2 \cdot 1}

u = \frac{- 1 + 5}{2 \cdot 1} : \frac{- 1 + 5}{2}

\frac{- 1 + 5}{2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{- 1 + 5}{2}

u = \frac{- 1 - 5}{2 \cdot 1} : \frac{- 1 - 5}{2}

\frac{- 1 - 5}{2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{- 1 - 5}{2}

Las soluciones a la ecuación de segundo grado son:

u = \frac{- 1 + 5}{2}, u = \frac{- 1 - 5}{2}

Sustituir en la ecuación

u = sin (2 x)

sin (2 x) = \frac{- 1 + 5}{2}, sin (2 x) = \frac{- 1 - 5}{2}

sin (2 x) = \frac{- 1 + 5}{2}, sin (2 x) = \frac{- 1 - 5}{2}

sin (2 x) = \frac{- 1 + 5}{2}, 0 \leq x \leq π : x = \frac{arcsin ( \frac{5 - 1}{2} )}{2}, x = \frac{π - arcsin ( \frac{5 - 1}{2} )}{2}

sin (2 x) = \frac{- 1 + 5}{2}, 0 \leq x \leq π

Aplicar propiedades trigonométricas inversas

sin (2 x) = \frac{- 1 + 5}{2}

Soluciones generales para

sin (2 x) = \frac{- 1 + 5}{2}

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π - arcsin (a) + 2 πn

2 x = arcsin (\frac{- 1 + 5}{2}) + 2 πn, 2 x = π - arcsin (\frac{- 1 + 5}{2}) + 2 πn

2 x = arcsin (\frac{- 1 + 5}{2}) + 2 πn, 2 x = π - arcsin (\frac{- 1 + 5}{2}) + 2 πn

Resolver

2 x = arcsin (\frac{- 1 + 5}{2}) + 2 πn : x = \frac{arcsin ( \frac{- 1 + 5}{2} )}{2} + πn

2 x = arcsin (\frac{- 1 + 5}{2}) + 2 πn

Dividir ambos lados entre

2

2 x = arcsin (\frac{- 1 + 5}{2}) + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{arcsin ( \frac{- 1 + 5}{2} )}{2} + \frac{2 πn}{2}

Simplificar

x = \frac{arcsin ( \frac{- 1 + 5}{2} )}{2} + πn

x = \frac{arcsin ( \frac{- 1 + 5}{2} )}{2} + πn

Resolver

2 x = π - arcsin (\frac{- 1 + 5}{2}) + 2 πn : x = \frac{π}{2} - \frac{arcsin ( \frac{- 1 + 5}{2} )}{2} + πn

2 x = π - arcsin (\frac{- 1 + 5}{2}) + 2 πn

Dividir ambos lados entre

2

2 x = π - arcsin (\frac{- 1 + 5}{2}) + 2 πn

Dividir ambos lados entre

2

\frac{2 x}{2} = \frac{π}{2} - \frac{arcsin ( \frac{- 1 + 5}{2} )}{2} + \frac{2 πn}{2}

Simplificar

x = \frac{π}{2} - \frac{arcsin ( \frac{- 1 + 5}{2} )}{2} + πn

x = \frac{π}{2} - \frac{arcsin ( \frac{- 1 + 5}{2} )}{2} + πn

x = \frac{arcsin ( \frac{- 1 + 5}{2} )}{2} + πn, x = \frac{π}{2} - \frac{arcsin ( \frac{- 1 + 5}{2} )}{2} + πn

Soluciones para el rango

0 \leq x \leq π

x = \frac{arcsin ( \frac{5 - 1}{2} )}{2}, x = \frac{π - arcsin ( \frac{5 - 1}{2} )}{2}

sin (2 x) = \frac{- 1 - 5}{2}, 0 \leq x \leq π :

Sin solución

sin (2 x) = \frac{- 1 - 5}{2}, 0 \leq x \leq π

- 1 \leq sin (x) \leq 1

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

x = \frac{arcsin ( \frac{5 - 1}{2} )}{2}, x = \frac{π - arcsin ( \frac{5 - 1}{2} )}{2}

Combinar toda las soluciones

x = \frac{π + arcsin ( \frac{5 - 1}{2} )}{2}, x = \frac{- arcsin ( \frac{5 - 1}{2} ) + 2 π}{2}, x = \frac{arcsin ( \frac{5 - 1}{2} )}{2}, x = \frac{π - arcsin ( \frac{5 - 1}{2} )}{2}

Verificar las soluciones sustituyendo en la ecuación original

Verificar las soluciones sustituyéndolas en

tan (2 x) = cos (2 x)

Quitar las que no concuerden con la ecuación.

Verificar la solución

\frac{π + arcsin ( \frac{5 - 1}{2} )}{2} :

Falso

\frac{π + arcsin ( \frac{5 - 1}{2} )}{2}

Sustituir

n = 1

\frac{π + arcsin ( \frac{5 - 1}{2} )}{2}

Multiplicar

tan (2 x) = cos (2 x)

por

x = \frac{π + arcsin ( \frac{5 - 1}{2} )}{2}

tan 2 \cdot \frac{π + arcsin ( \frac{5 - 1}{2} )}{2} = cos 2 \cdot \frac{π + arcsin ( \frac{5 - 1}{2} )}{2}

Simplificar

0.78615 \dots = - 0.78615 \dots

\Rightarrow Falso

Verificar la solución

\frac{- arcsin ( \frac{5 - 1}{2} ) + 2 π}{2} :

Falso

\frac{- arcsin ( \frac{5 - 1}{2} ) + 2 π}{2}

Sustituir

n = 1

\frac{- arcsin ( \frac{5 - 1}{2} ) + 2 π}{2}

Multiplicar

tan (2 x) = cos (2 x)

por

x = \frac{- arcsin ( \frac{5 - 1}{2} ) + 2 π}{2}

tan 2 \cdot \frac{- arcsin ( \frac{5 - 1}{2} ) + 2 π}{2} = cos 2 \cdot \frac{- arcsin ( \frac{5 - 1}{2} ) + 2 π}{2}

Simplificar

- 0.78615 \dots = 0.78615 \dots

\Rightarrow Falso

Verificar la solución

\frac{arcsin ( \frac{5 - 1}{2} )}{2} :

Verdadero

\frac{arcsin ( \frac{5 - 1}{2} )}{2}

Sustituir

n = 1

\frac{arcsin ( \frac{5 - 1}{2} )}{2}

Multiplicar

tan (2 x) = cos (2 x)

por

x = \frac{arcsin ( \frac{5 - 1}{2} )}{2}

tan 2 \cdot \frac{arcsin ( \frac{5 - 1}{2} )}{2} = cos 2 \cdot \frac{arcsin ( \frac{5 - 1}{2} )}{2}

Simplificar

0.78615 \dots = 0.78615 \dots

\Rightarrow Verdadero

Verificar la solución

\frac{π - arcsin ( \frac{5 - 1}{2} )}{2} :

Verdadero

\frac{π - arcsin ( \frac{5 - 1}{2} )}{2}

Sustituir

n = 1

\frac{π - arcsin ( \frac{5 - 1}{2} )}{2}

Multiplicar

tan (2 x) = cos (2 x)

por

x = \frac{π - arcsin ( \frac{5 - 1}{2} )}{2}

tan 2 \cdot \frac{π - arcsin ( \frac{5 - 1}{2} )}{2} = cos 2 \cdot \frac{π - arcsin ( \frac{5 - 1}{2} )}{2}

Simplificar

- 0.78615 \dots = - 0.78615 \dots

\Rightarrow Verdadero

x = \frac{arcsin ( \frac{5 - 1}{2} )}{2}, x = \frac{π - arcsin ( \frac{5 - 1}{2} )}{2}

Mostrar soluciones en forma decimal

x = \frac{0.66623 \dots}{2}, x = \frac{π - 0.66623 \dots}{2}

Ejemplos populares

5sin(2x)=5cos(x),0<= x<= 2pi

5 sin (2 x) = 5 cos (x), 0 \leq x \leq 2 π

solvefor x,f=cos(x)cos(hy)resolver para

x, f = cos (x) cos (h y)

tan(x)=-1/10

tan (x) = - \frac{1}{10}

tan(4x)*cot(x+60)=1

tan (4 x) \cdot cot (x + 60) = 1

sin^2(A)+cos^2(A)+sin(A)-2=0

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