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Popular >

(50)/(sin(105))=(20)/(sin(x))

Solución

\frac{50}{sin ( 10 5 ^{\circ} )} = \frac{20}{sin ( x )}

Solución

x = 0.39669 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.39669 \dots + 36 0^{\circ} n

radianes

x = 0.39669 \dots + 2 πn, x = π - 0.39669 \dots + 2 πn

Pasos de solución

\frac{50}{sin ( 10 5 ^{\circ} )} = \frac{20}{sin ( x )}

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{50}{\frac{2 ( 3 + 1 )}{4}} = \frac{20}{sin ( x )}

Utilizar multiplicación cruzada (regla de tres)

\frac{50}{\frac{2 ( 3 + 1 )}{4}} = \frac{20}{sin ( x )}

Utilizar multiplicación cruzada (regla de tres): Si

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

entonces

a \cdot d = b \cdot c

50 sin (x) = \frac{2 ( 3 + 1 )}{4} \cdot 20

Simplificar

\frac{2 ( 3 + 1 )}{4} \cdot 20 : 52 (1 + 3)

\frac{2 ( 3 + 1 )}{4} \cdot 20

Multiplicar fracciones:

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

= \frac{2 ( 3 + 1 ) \cdot 20}{4}

Dividir:

\frac{20}{4} = 5

= 52 (1 + 3)

50 sin (x) = 52 (1 + 3)

50 sin (x) = 52 (1 + 3)

Dividir ambos lados entre

50

50 sin (x) = 52 (1 + 3)

Dividir ambos lados entre

50

\frac{50 sin ( x )}{50} = \frac{5 2 ( 1 + 3 )}{50}

Simplificar

sin (x) = \frac{2 ( 1 + 3 )}{10}

sin (x) = \frac{2 ( 1 + 3 )}{10}

Verificar las soluciones

Encontrar los puntos no definidos (singularidades):

sin (x) = 0

Tomar el(los) denominador(es) de

\frac{20}{sin ( x )}

y comparar con cero

sin (x) = 0

Los siguientes puntos no están definidos

sin (x) = 0

Combinar los puntos no definidos con las soluciones:

sin (x) = \frac{2 ( 1 + 3 )}{10}

Aplicar propiedades trigonométricas inversas

sin (x) = \frac{2 ( 1 + 3 )}{10}

Soluciones generales para

sin (x) = \frac{2 ( 1 + 3 )}{10}

sin (x) = a \Rightarrow x = arcsin (a) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (a) + 36 0^{\circ} n

x = arcsin (\frac{2 ( 1 + 3 )}{10}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{2 ( 1 + 3 )}{10}) + 36 0^{\circ} n

x = arcsin (\frac{2 ( 1 + 3 )}{10}) + 36 0^{\circ} n, x = 18 0^{\circ} - arcsin (\frac{2 ( 1 + 3 )}{10}) + 36 0^{\circ} n

Mostrar soluciones en forma decimal

x = 0.39669 \dots + 36 0^{\circ} n, x = 18 0^{\circ} - 0.39669 \dots + 36 0^{\circ} n

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