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Popular >

(100sin(30))/(sin(105))

Solución

\frac{100 sin ( 3 0 ^{\circ} )}{sin ( 10 5 ^{\circ} )}

Solución

502 (3 - 1)

Notación decimal

51.76380 \dots

Pasos de solución

\frac{100 sin ( 3 0 ^{\circ} )}{sin ( 10 5 ^{\circ} )}

Utilizar la siguiente identidad trivial:

sin (3 0^{\circ}) = \frac{1}{2}

sin (3 0^{\circ})

tabla de valores periódicos con

36 0^{\circ} n

intervalos:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{1}{2}

Re-escribir usando identidades trigonométricas:

sin (10 5^{\circ}) = \frac{2 ( 3 + 1 )}{4}

sin (10 5^{\circ})

Re-escribir usando identidades trigonométricas:

sin (6 0^{\circ}) cos (4 5^{\circ}) + cos (6 0^{\circ}) sin (4 5^{\circ})

sin (10 5^{\circ})

Escribir

sin (10 5^{\circ})

como

sin (6 0^{\circ} + 4 5^{\circ})

= sin (6 0^{\circ} + 4 5^{\circ})

Utilizar la identidad de suma de ángulos:

sin (s + t) = sin (s) cos (t) + cos (s) sin (t)

= sin (6 0^{\circ}) cos (4 5^{\circ}) + cos (6 0^{\circ}) sin (4 5^{\circ})

= sin (6 0^{\circ}) cos (4 5^{\circ}) + cos (6 0^{\circ}) sin (4 5^{\circ})

Utilizar la siguiente identidad trivial:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 0^{\circ})

tabla de valores periódicos con

36 0^{\circ} n

intervalos:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

Utilizar la siguiente identidad trivial:

cos (4 5^{\circ}) = \frac{2}{2}

cos (4 5^{\circ})

cos (x)

tabla de valores periódicos con

36 0^{\circ} n

intervalos:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

Utilizar la siguiente identidad trivial:

cos (6 0^{\circ}) = \frac{1}{2}

cos (6 0^{\circ})

cos (x)

tabla de valores periódicos con

36 0^{\circ} n

intervalos:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{1}{2}

Utilizar la siguiente identidad trivial:

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

sin (4 5^{\circ})

tabla de valores periódicos con

36 0^{\circ} n

intervalos:

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

= \frac{3}{2} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{2}{2}

Simplificar

\frac{3}{2} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{2}{2} : \frac{2 ( 3 + 1 )}{4}

\frac{3}{2} \cdot \frac{2}{2} + \frac{1}{2} \cdot \frac{2}{2}

Factorizar el termino común

\frac{2}{2}

= \frac{2}{2} (\frac{3}{2} + \frac{1}{2})

\frac{3}{2} + \frac{1}{2} = \frac{3 + 1}{2}

\frac{3}{2} + \frac{1}{2}

Aplicar la regla

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

= \frac{3 + 1}{2}

= \frac{2}{2} \cdot \frac{1 + 3}{2}

Multiplicar fracciones:

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

= \frac{( 3 + 1 ) 2}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{2 ( 1 + 3 )}{4}

= \frac{2 ( 3 + 1 )}{4}

= \frac{100 \cdot \frac{1}{2}}{\frac{2 ( 3 + 1 )}{4}}

Simplificar

\frac{100 \cdot \frac{1}{2}}{\frac{2 ( 3 + 1 )}{4}} : 502 (3 - 1)

\frac{100 \cdot \frac{1}{2}}{\frac{2 ( 3 + 1 )}{4}}

Aplicar las propiedades de las fracciones:

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

= \frac{100 \cdot \frac{1}{2} \cdot 4}{2 ( 3 + 1 )}

Multiplicar los numeros:

100 \cdot 4 = 400

= \frac{400 \cdot \frac{1}{2}}{2 ( 1 + 3 )}

Multiplicar

400 \cdot \frac{1}{2} : 200

400 \cdot \frac{1}{2}

Multiplicar fracciones:

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

= \frac{1 \cdot 400}{2}

Multiplicar los numeros:

1 \cdot 400 = 400

= \frac{400}{2}

Dividir:

\frac{400}{2} = 200

= 200

= \frac{200}{2 ( 1 + 3 )}

Factorizar

200 : 2^{3} \cdot 5^{2}

Factorizar

200 = 2^{3} \cdot 5^{2}

= \frac{2 ^{3} \cdot 5 ^{2}}{2 ( 1 + 3 )}

Cancelar

\frac{2 ^{3} \cdot 5 ^{2}}{2 ( 3 + 1 )} : \frac{5 ^{2} \cdot 2 ^{\frac{5}{2}}}{3 + 1}

\frac{2 ^{3} \cdot 5 ^{2}}{2 ( 3 + 1 )}

Aplicar las leyes de los exponentes:

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

= \frac{2 ^{3} \cdot 5 ^{2}}{2 ^{\frac{1}{2}} ( 1 + 3 )}

Aplicar las leyes de los exponentes:

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

\frac{2 ^{3}}{2 ^{\frac{1}{2}}} = 2^{3 - \frac{1}{2}}

= \frac{5 ^{2} \cdot 2 ^{- \frac{1}{2} + 3}}{3 + 1}

Restar:

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

= \frac{5 ^{2} \cdot 2 ^{\frac{5}{2}}}{3 + 1}

= \frac{5 ^{2} \cdot 2 ^{\frac{5}{2}}}{3 + 1}

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

2^{\frac{5}{2}}

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

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

Aplicar las leyes de los exponentes:

x^{a + b} = x^{a} x^{b}

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

Simplificar

= 2^{2} 2

= \frac{5 ^{2} \cdot 2 ^{2} 2}{3 + 1}

5^{2} \cdot 2^{2} 2 = 1002

5^{2} \cdot 2^{2} 2

5^{2} = 25

= 2^{2} \cdot 252

2^{2} = 4

= 25 \cdot 42

Multiplicar los numeros:

25 \cdot 4 = 100

= 1002

= \frac{100 2}{3 + 1}

Racionalizar

\frac{100 2}{3 + 1} : 502 (3 - 1)

\frac{100 2}{3 + 1}

Multiplicar por el conjugado

\frac{3 - 1}{3 - 1}

= \frac{100 2 ( 3 - 1 )}{( 3 + 1 ) ( 3 - 1 )}

(3 + 1) (3 - 1) = 2

(3 + 1) (3 - 1)

Aplicar la siguiente regla para binomios al cuadrado:

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

a = 3, b = 1

= (3)^{2} - 1^{2}

Simplificar

(3)^{2} - 1^{2} : 2

(3)^{2} - 1^{2}

Aplicar la regla

1^{a} = 1

1^{2} = 1

= (3)^{2} - 1

(3)^{2} = 3

(3)^{2}

Aplicar las leyes de los exponentes:

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

= (3^{\frac{1}{2}})^{2}

Aplicar las leyes de los exponentes:

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

= 3^{\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

= 3

= 3 - 1

Restar:

3 - 1 = 2

= 2

= 2

= \frac{100 2 ( 3 - 1 )}{2}

Dividir:

\frac{100}{2} = 50

= 502 (3 - 1)

= 502 (3 - 1)

= 502 (3 - 1)

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