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(((1-sin^2(x)))/((1+2*sin^2(x))))=0.45

Solución

(\frac{( 1 - sin ^{2} ( x ) )}{( 1 + 2 \cdot sin ^{2} ( x ) )}) = 0.45

Solución

x = 0.56809 \dots + 2 πn, x = π - 0.56809 \dots + 2 πn, x = - 0.56809 \dots + 2 πn, x = π + 0.56809 \dots + 2 πn

Grados

x = 32.54946 \dots^{\circ} + 36 0^{\circ} n, x = 147.45053 \dots^{\circ} + 36 0^{\circ} n, x = - 32.54946 \dots^{\circ} + 36 0^{\circ} n, x = 212.54946 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

(\frac{( 1 - sin ^{2} ( x ) )}{( 1 + 2 sin ^{2} ( x ) )}) = 0.45

Usando el método de sustitución

\frac{1 - sin ^{2} ( x )}{1 + 2 sin ^{2} ( x )} = 0.45

Sea:

sin (x) = u

\frac{1 - u ^{2}}{1 + 2 u ^{2}} = 0.45

\frac{1 - u ^{2}}{1 + 2 u ^{2}} = 0.45 : u = \frac{11}{38}, u = - \frac{11}{38}

\frac{1 - u ^{2}}{1 + 2 u ^{2}} = 0.45

Multiplicar ambos lados por

1 + 2 u^{2}

\frac{1 - u ^{2}}{1 + 2 u ^{2}} = 0.45

Multiplicar ambos lados por

1 + 2 u^{2}

\frac{1 - u ^{2}}{1 + 2 u ^{2}} (1 + 2 u^{2}) = 0.45 (1 + 2 u^{2})

Simplificar

1 - u^{2} = 0.45 (1 + 2 u^{2})

1 - u^{2} = 0.45 (1 + 2 u^{2})

Resolver

1 - u^{2} = 0.45 (1 + 2 u^{2}) : u = \frac{11}{38}, u = - \frac{11}{38}

1 - u^{2} = 0.45 (1 + 2 u^{2})

Desarrollar

0.45 (1 + 2 u^{2}) : 0.45 + 0.9 u^{2}

0.45 (1 + 2 u^{2})

Poner los parentesis utilizando:

a (b + c) = ab + a c

a = 0.45, b = 1, c = 2 u^{2}

= 0.45 \cdot 1 + 0.45 \cdot 2 u^{2}

= 1 \cdot 0.45 + 2 \cdot 0.45 u^{2}

Simplificar

1 \cdot 0.45 + 2 \cdot 0.45 u^{2} : 0.45 + 0.9 u^{2}

1 \cdot 0.45 + 2 \cdot 0.45 u^{2}

Multiplicar los numeros:

1 \cdot 0.45 = 0.45

= 0.45 + 2 \cdot 0.45 u^{2}

Multiplicar los numeros:

2 \cdot 0.45 = 0.9

= 0.45 + 0.9 u^{2}

= 0.45 + 0.9 u^{2}

1 - u^{2} = 0.45 + 0.9 u^{2}

Mover

1

al lado derecho

1 - u^{2} = 0.45 + 0.9 u^{2}

Restar

1

de ambos lados

1 - u^{2} - 1 = 0.45 + 0.9 u^{2} - 1

Simplificar

- u^{2} = 0.9 u^{2} - 0.55

- u^{2} = 0.9 u^{2} - 0.55

Mover

0.9 u^{2}

al lado izquierdo

- u^{2} = 0.9 u^{2} - 0.55

Restar

0.9 u^{2}

de ambos lados

- u^{2} - 0.9 u^{2} = 0.9 u^{2} - 0.55 - 0.9 u^{2}

Simplificar

- 1.9 u^{2} = - 0.55

- 1.9 u^{2} = - 0.55

Multiplicar ambos lados por

100

- 1.9 u^{2} = - 0.55

To eliminate decimal points, multiply by 10 for every digit after the decimal pointThere are

2

digits to the right of the decimal point, therefore multiply by

100

- 1.9 u^{2} \cdot 100 = - 0.55 \cdot 100

Simplificar

- 190 u^{2} = - 55

- 190 u^{2} = - 55

Dividir ambos lados entre

- 190

- 190 u^{2} = - 55

Dividir ambos lados entre

- 190

\frac{- 190 u ^{2}}{- 190} = \frac{- 55}{- 190}

Simplificar

\frac{- 190 u ^{2}}{- 190} = \frac{- 55}{- 190}

Simplificar

\frac{- 190 u ^{2}}{- 190} : u^{2}

\frac{- 190 u ^{2}}{- 190}

Aplicar las propiedades de las fracciones:

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

= \frac{190 u ^{2}}{190}

Dividir:

\frac{190}{190} = 1

= u^{2}

Simplificar

\frac{- 55}{- 190} : \frac{11}{38}

\frac{- 55}{- 190}

Aplicar las propiedades de las fracciones:

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

= \frac{55}{190}

Eliminar los terminos comunes:

5

= \frac{11}{38}

u^{2} = \frac{11}{38}

u^{2} = \frac{11}{38}

u^{2} = \frac{11}{38}

Para

x^{2} = f (a)

las soluciones son

x = f (a), - f (a)

u = \frac{11}{38}, u = - \frac{11}{38}

u = \frac{11}{38}, u = - \frac{11}{38}

Sustituir en la ecuación

u = sin (x)

sin (x) = \frac{11}{38}, sin (x) = - \frac{11}{38}

sin (x) = \frac{11}{38}, sin (x) = - \frac{11}{38}

sin (x) = \frac{11}{38} : x = arcsin (\frac{11}{38}) + 2 πn, x = π - arcsin (\frac{11}{38}) + 2 πn

sin (x) = \frac{11}{38}

Aplicar propiedades trigonométricas inversas

sin (x) = \frac{11}{38}

Soluciones generales para

sin (x) = \frac{11}{38}

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

x = arcsin (\frac{11}{38}) + 2 πn, x = π - arcsin (\frac{11}{38}) + 2 πn

x = arcsin (\frac{11}{38}) + 2 πn, x = π - arcsin (\frac{11}{38}) + 2 πn

sin (x) = - \frac{11}{38} : x = arcsin (- \frac{11}{38}) + 2 πn, x = π + arcsin (\frac{11}{38}) + 2 πn

sin (x) = - \frac{11}{38}

Aplicar propiedades trigonométricas inversas

sin (x) = - \frac{11}{38}

Soluciones generales para

sin (x) = - \frac{11}{38}

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

x = arcsin (- \frac{11}{38}) + 2 πn, x = π + arcsin (\frac{11}{38}) + 2 πn

x = arcsin (- \frac{11}{38}) + 2 πn, x = π + arcsin (\frac{11}{38}) + 2 πn

Combinar toda las soluciones

x = arcsin (\frac{11}{38}) + 2 πn, x = π - arcsin (\frac{11}{38}) + 2 πn, x = arcsin (- \frac{11}{38}) + 2 πn, x = π + arcsin (\frac{11}{38}) + 2 πn

Mostrar soluciones en forma decimal

x = 0.56809 \dots + 2 πn, x = π - 0.56809 \dots + 2 πn, x = - 0.56809 \dots + 2 πn, x = π + 0.56809 \dots + 2 πn

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