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sqrt(1-sin^2(x))=e^{1/2 ln(1-cos^2(x))}

Solución

1 - sin^{2} (x) = e^{\frac{1}{2} l n (1 - c o s^{2} (x))}

Solución

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Grados

x = 4 5^{\circ} + 36 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Pasos de solución

1 - sin^{2} (x) = e^{\frac{1}{2} l n (1 - c o s^{2} (x))}

Restar

e^{\frac{1}{2} l n (1 - c o s^{2} (x))}

de ambos lados

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

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

Usando el método de sustitución

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

Sea:

sin (x) = u

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

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

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

Eliminar raíces cuadradas

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

Sumar

u^{2}

a ambos lados

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

Simplificar

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

Elevar al cuadrado ambos lados:

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

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

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

Desarrollar

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar las leyes de los exponentes:

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

= (1 - u^{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

= 1 - u^{2}

Desarrollar

(u^{2})^{2} : u^{2}

(u^{2})^{2}

Aplicar las leyes de los exponentes:

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

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

Aplicar las leyes de los exponentes:

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

= (u^{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

= u^{2}

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

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

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

Resolver

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

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

Mover

1

al lado derecho

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

Restar

1

de ambos lados

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

Simplificar

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

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

Mover

u^{2}

al lado izquierdo

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

Restar

u^{2}

de ambos lados

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

Simplificar

- 2 u^{2} = - 1

- 2 u^{2} = - 1

Dividir ambos lados entre

- 2

- 2 u^{2} = - 1

Dividir ambos lados entre

- 2

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

Simplificar

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

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

Para

x^{2} = f (a)

las soluciones son

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

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

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

Verificar las soluciones:

u = \frac{1}{2}

Verdadero

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

Verdadero

Verificar las soluciones sustituyéndolas en

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

Quitar las que no concuerden con la ecuación.

Sustituir

u = \frac{1}{2} :

Verdadero

1 - (\frac{1}{2})^{2} - (\frac{1}{2})^{2} = 0

1 - (\frac{1}{2})^{2} - (\frac{1}{2})^{2} = 0

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

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

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

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar las leyes de los exponentes:

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

= (\frac{1}{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

= \frac{1}{2}

= 1 - \frac{1}{2}

Simplificar

1 - \frac{1}{2}

en una fracción:

\frac{1}{2}

1 - \frac{1}{2}

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

1 \cdot 2 - 1 = 1

1 \cdot 2 - 1

Multiplicar los numeros:

1 \cdot 2 = 2

= 2 - 1

Restar:

2 - 1 = 1

= 1

= \frac{1}{2}

= \frac{1}{2}

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

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

Aplicar la siguiente propiedad de los radicales:

asumiendo que

a \geq 0

= \frac{1}{2}

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

Sumar elementos similares:

\frac{1}{2} - \frac{1}{2} = 0

= 0

0 = 0

Verdadero

Sustituir

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

Verdadero

1 - (- \frac{1}{2})^{2} - (- \frac{1}{2})^{2} = 0

1 - (- \frac{1}{2})^{2} - (- \frac{1}{2})^{2} = 0

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

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

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

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

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

Aplicar las leyes de los exponentes:

(- a)^{n} = a^{n},

n

es par

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar las leyes de los exponentes:

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

= (\frac{1}{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

= \frac{1}{2}

= 1 - \frac{1}{2}

Simplificar

1 - \frac{1}{2}

en una fracción:

\frac{1}{2}

1 - \frac{1}{2}

Convertir a fracción:

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

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

Ya que los denominadores son iguales, combinar las fracciones:

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

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

1 \cdot 2 - 1 = 1

1 \cdot 2 - 1

Multiplicar los numeros:

1 \cdot 2 = 2

= 2 - 1

Restar:

2 - 1 = 1

= 1

= \frac{1}{2}

= \frac{1}{2}

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

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

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

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

Aplicar las leyes de los exponentes:

(- a)^{n} = a^{n},

n

es par

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

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

Aplicar las leyes de los exponentes:

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

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

Aplicar las leyes de los exponentes:

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

= (\frac{1}{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

= \frac{1}{2}

= \frac{1}{2}

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

Sumar elementos similares:

\frac{1}{2} - \frac{1}{2} = 0

= 0

0 = 0

Verdadero

Las soluciones son

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

Sustituir en la ecuación

u = sin (x)

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

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

sin (x) = \frac{1}{2} : x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = \frac{1}{2}

Soluciones generales para

sin (x) = \frac{1}{2}

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn

sin (x) = - \frac{1}{2} : x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

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

Soluciones generales para

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

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Combinar toda las soluciones

x = \frac{π}{4} + 2 πn, x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

Ejemplos populares

cot^2(x)=tan(1/2 x)

cot^{2} (x) = tan (\frac{1}{2} x)

cos(C)= 3/4

cos (C) = \frac{3}{4}

cos(x-pi/6)-sin(x)=0

cos (x - \frac{π}{6}) - sin (x) = 0

2+cos^2(x)=-3cos(x)

2 + cos^{2} (x) = - 3 cos (x)

6sec^2(x)+tan(x)-7=0

6 sec^{2} (x) + tan (x) - 7 = 0