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Popular >

(8.2)/(cos(72))

Solución

\frac{8.2}{cos ( 7 2 ^{\circ} )}

Solución

\frac{41 ( 1 + 5 )}{5}

Notación decimal

26.53575 \dots

Pasos de solución

\frac{8.2}{cos ( 7 2 ^{\circ} )}

= \frac{\frac{41}{5}}{cos ( 7 2 ^{\circ} )}

Simplificar

= \frac{41}{5 cos ( 7 2 ^{\circ} )}

Re-escribir usando identidades trigonométricas:

cos (7 2^{\circ}) = 1 - 2 sin^{2} (3 6^{\circ})

cos (7 2^{\circ})

Escribir

cos (7 2^{\circ})

como

cos (2 \cdot 3 6^{\circ})

= cos (2 \cdot 3 6^{\circ})

Utilizar la identidad trigonométrica del ángulo doble:

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

= 1 - 2 sin^{2} (3 6^{\circ})

= \frac{41}{5 ( 1 - 2 sin ^{2} ( 3 6 ^{\circ} ) )}

Re-escribir usando identidades trigonométricas:

sin (3 6^{\circ}) = \frac{2 5 - 5}{4}

sin (3 6^{\circ})

Demostrar que:

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

Utilizar el siguiente producto para la identidad de suma de ángulos:

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

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = sin (5 4^{\circ}) - sin (1 8^{\circ})

Demostrar que:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

Utilizar la identidad trigonométrica del ángulo doble:

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

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Dividir ambos lados entre

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Usar la siguiente identidad:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Dividir ambos lados entre

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

Dividir ambos lados entre

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

Sustituir

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

\frac{1}{2} = sin (5 4^{\circ}) - sin (1 8^{\circ})

sin (5 4^{\circ}) = cos (9 0^{\circ} - 5 4^{\circ})

\frac{1}{2} = cos (9 0^{\circ} - 5 4^{\circ}) - sin (1 8^{\circ})

\frac{1}{2} = cos (3 6^{\circ}) - sin (1 8^{\circ})

Demostrar que:

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

Utilizar la regla de factorización:

a^{2} - b^{2} = (a + b) (a - b)

a = cos (3 6^{\circ}) + sin (1 8^{\circ})

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = ((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) ((cos (3 6^{\circ}) + sin (1 8^{\circ})) - (cos (3 6^{\circ}) - sin (1 8^{\circ})))

Simplificar

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 2 (2 cos (3 6^{\circ}) sin (1 8^{\circ}))

Demostrar que:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

Utilizar la identidad trigonométrica del ángulo doble:

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

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Dividir ambos lados entre

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Usar la siguiente identidad:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

Dividir ambos lados entre

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

Dividir ambos lados entre

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

Sustituir

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 1

Sustituir

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (\frac{1}{2})^{2} = 1

Simplificar

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} = 1

Sumar

\frac{1}{4}

a ambos lados

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} + \frac{1}{4} = 1 + \frac{1}{4}

Simplificar

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} = \frac{5}{4}

Obtener la raíz cuadrada de ambos lados

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \pm \frac{5}{4}

cos (3 6^{\circ})

no puede ser negativa

sin (1 8^{\circ})

no puede ser negativa

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

Añadir las siguientes ecuaciones

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{2}

((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) = (\frac{5}{2} + \frac{1}{2})

Simplificar

cos (3 6^{\circ}) = \frac{5 + 1}{4}

Elevar al cuadrado ambos lados

(cos (3 6^{\circ}))^{2} = (\frac{5 + 1}{4})^{2}

Usar la siguiente identidad:

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

sin^{2} (3 6^{\circ}) = 1 - cos^{2} (3 6^{\circ})

Sustituir

cos (3 6^{\circ}) = \frac{5 + 1}{4}

sin^{2} (3 6^{\circ}) = 1 - (\frac{5 + 1}{4})^{2}

Simplificar

sin^{2} (3 6^{\circ}) = \frac{5 - 5}{8}

Obtener la raíz cuadrada de ambos lados

sin (3 6^{\circ}) = \pm \frac{5 - 5}{8}

sin (3 6^{\circ})

no puede ser negativa

sin (3 6^{\circ}) = \frac{5 - 5}{8}

Simplificar

sin (3 6^{\circ}) = \frac{\frac{5 - 5}{2}}{2}

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

Simplificar

= \frac{2 5 - 5}{4}

= \frac{41}{5 ( 1 - 2 ( \frac{2 5 - 5}{4} ) ^{2} )}

Simplificar

\frac{41}{5 ( 1 - 2 ( \frac{2 5 - 5}{4} ) ^{2} )} : \frac{41 ( 1 + 5 )}{5}

\frac{41}{5 ( 1 - 2 ( \frac{2 5 - 5}{4} ) ^{2} )}

2 (\frac{2 5 - 5}{4})^{2} = \frac{5 - 5}{4}

2 (\frac{2 5 - 5}{4})^{2}

(\frac{2 5 - 5}{4})^{2} = \frac{5 - 5}{2 ^{3}}

(\frac{2 5 - 5}{4})^{2}

Aplicar las leyes de los exponentes:

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

= \frac{( 2 5 - 5 ) ^{2}}{4 ^{2}}

Aplicar las leyes de los exponentes:

(a \cdot b)^{n} = a^{n} b^{n}

(2 5 - 5)^{2} = (2)^{2} (5 - 5)^{2}

= \frac{( 2 ) ^{2} ( 5 - 5 ) ^{2}}{4 ^{2}}

(2)^{2} : 2

Aplicar las leyes de los exponentes:

a = a^{\frac{1}{2}}

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

Aplicar las leyes de los exponentes:

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

= 2^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

2

= 1

= 2

= \frac{2 ( 5 - 5 ) ^{2}}{4 ^{2}}

(5 - 5)^{2} : 5 - 5

Aplicar las leyes de los exponentes:

a = a^{\frac{1}{2}}

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

Aplicar las leyes de los exponentes:

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

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

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

\frac{1}{2} \cdot 2

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

2

= 1

= 5 - 5

= \frac{2 ( 5 - 5 )}{4 ^{2}}

Factorizar

4^{2} : 2^{4}

Factorizar

4 = 2^{2}

= (2^{2})^{2}

Simplificar

(2^{2})^{2} : 2^{4}

(2^{2})^{2}

Aplicar las leyes de los exponentes:

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

= 2^{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= 2^{4}

= 2^{4}

= \frac{2 ( 5 - 5 )}{2 ^{4}}

Eliminar los terminos comunes:

2

= \frac{5 - 5}{2 ^{3}}

= 2 \cdot \frac{5 - 5}{2 ^{3}}

Multiplicar fracciones:

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

= \frac{( 5 - 5 ) \cdot 2}{2 ^{3}}

Eliminar los terminos comunes:

2

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

2^{2} = 4

= \frac{5 - 5}{4}

= \frac{41}{5 ( - \frac{5 - 5}{4} + 1 )}

Simplificar

1 - \frac{5 - 5}{4}

en una fracción:

\frac{5 - 1}{4}

1 - \frac{5 - 5}{4}

Convertir a fracción:

1 = \frac{1 \cdot 4}{4}

= \frac{1 \cdot 4}{4} - \frac{5 - 5}{4}

Ya que los denominadores son iguales, combinar las fracciones:

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

= \frac{1 \cdot 4 - ( 5 - 5 )}{4}

Multiplicar los numeros:

1 \cdot 4 = 4

= \frac{4 - ( 5 - 5 )}{4}

Expandir

4 - (5 - 5) : 5 - 1

4 - (5 - 5)

- (5 - 5) : - 5 + 5

- (5 - 5)

Poner los parentesis

= - (5) - (- 5)

Aplicar las reglas de los signos

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

= - 5 + 5

= 4 - 5 + 5

Restar:

4 - 5 = - 1

= 5 - 1

= \frac{5 - 1}{4}

= \frac{41}{5 \cdot \frac{5 - 1}{4}}

Multiplicar

5 \cdot \frac{5 - 1}{4} : \frac{5 ( 5 - 1 )}{4}

5 \cdot \frac{5 - 1}{4}

Multiplicar fracciones:

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

= \frac{( 5 - 1 ) \cdot 5}{4}

= \frac{41}{\frac{5 ( 5 - 1 )}{4}}

Aplicar las propiedades de las fracciones:

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

= \frac{41 \cdot 4}{( 5 - 1 ) \cdot 5}

Multiplicar los numeros:

41 \cdot 4 = 164

= \frac{164}{5 ( 5 - 1 )}

Racionalizar

\frac{164}{5 ( 5 - 1 )} : \frac{41 ( 1 + 5 )}{5}

\frac{164}{5 ( 5 - 1 )}

Multiplicar por el conjugado

\frac{5 + 1}{5 + 1}

= \frac{164 ( 5 + 1 )}{( 5 - 1 ) \cdot 5 ( 5 + 1 )}

(5 - 1) \cdot 5 (5 + 1) = 20

(5 - 1) \cdot 5 (5 + 1)

= 5 (5 - 1) (5 + 1)

Expandir

(5 - 1) (5 + 1) : 4

(5 - 1) (5 + 1)

Aplicar la siguiente regla para binomios al cuadrado:

(a - b) (a + b) = a^{2} - b^{2}

a = 5, b = 1

= (5)^{2} - 1^{2}

Simplificar

(5)^{2} - 1^{2} : 4

(5)^{2} - 1^{2}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= (5)^{2} - 1

(5)^{2} = 5

(5)^{2}

Aplicar las leyes de los exponentes:

a = a^{\frac{1}{2}}

= (5^{\frac{1}{2}})^{2}

Aplicar las leyes de los exponentes:

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

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

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

\frac{1}{2} \cdot 2

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

2

= 1

= 5

= 5 - 1

Restar:

5 - 1 = 4

= 4

= 4

= 5 \cdot 4

Expandir

5 \cdot 4 : 20

5 \cdot 4

Aplicar la siguiente regla de productos notables

= 5 \cdot 4

Multiplicar los numeros:

5 \cdot 4 = 20

= 20

= 20

= \frac{164 ( 5 + 1 )}{20}

Eliminar los terminos comunes:

4

= \frac{41 ( 1 + 5 )}{5}

= \frac{41 ( 1 + 5 )}{5}

= \frac{41 ( 1 + 5 )}{5}

Ejemplos populares

sin(1/(-1))

sin (\frac{1}{- 1})

cos(2arctan(1/7)+arcsin(3/7))

cos (2 arctan (\frac{1}{7}) + arcsin (\frac{3}{7}))

sin(arctan((-12)/5))

sin (arctan (\frac{- 12}{5}))

2sin(0)cos(0)

2 sin (0) cos (0)

(tan(45))/(cos(45))

\frac{tan ( 4 5 ^{\circ} )}{cos ( 4 5 ^{\circ} )}