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Popular >

0=sqrt(((10cos(x))-8)^2((10sin(x))-6)^2)

Solución

0 = ((10 cos (x)) - 8)^{2} ((10 sin (x)) - 6)^{2}

Solución

x = 2 π - 0.64350 \dots + 2 πn, x = 0.64350 \dots + 2 πn, x = π - 0.64350 \dots + 2 πn

Grados

x = 323.13010 \dots^{\circ} + 36 0^{\circ} n, x = 36.86989 \dots^{\circ} + 36 0^{\circ} n, x = 143.13010 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

0 = ((10 cos (x)) - 8)^{2} ((10 sin (x)) - 6)^{2}

Intercambiar lados

(10 cos (x) - 8)^{2} (10 sin (x) - 6)^{2} = 0

Factorizar

(10 cos (x) - 8)^{2} (10 sin (x) - 6)^{2} : 4 (5 cos (x) - 4)^{2} (5 sin (x) - 3)^{2}

(10 cos (x) - 8)^{2} (10 sin (x) - 6)^{2}

(10 cos (x) - 8)^{2} = (2 (5 cos (x) - 4))^{2}

(10 cos (x) - 8)^{2}

Factorizar

10 cos (x) - 8 : 2 (5 cos (x) - 4)

10 cos (x) - 8

Factorizar el termino común

2 : 2 (5 cos (x) - 4)

10 cos (x) - 8

Reescribir

8

como

2 \cdot 4

Reescribir

10

como

2 \cdot 5

= 2 \cdot 5 cos (x) - 2 \cdot 4

Factorizar el termino común

2

= 2 (5 cos (x) - 4)

= 2 (5 cos (x) - 4)

= (2 (5 cos (x) - 4))^{2}

= (2 (5 cos (x) - 4))^{2} (10 sin (x) - 6)^{2}

(10 sin (x) - 6)^{2} = (2 (5 sin (x) - 3))^{2}

(10 sin (x) - 6)^{2}

Factorizar

10 sin (x) - 6 : 2 (5 sin (x) - 3)

10 sin (x) - 6

Factorizar el termino común

2 : 2 (5 sin (x) - 3)

10 sin (x) - 6

Reescribir

6

como

2 \cdot 3

Reescribir

10

como

2 \cdot 5

= 2 \cdot 5 sin (x) - 2 \cdot 3

Factorizar el termino común

2

= 2 (5 sin (x) - 3)

= 2 (5 sin (x) - 3)

= (2 (5 sin (x) - 3))^{2}

= (2 (5 cos (x) - 4))^{2} (2 (5 sin (x) - 3))^{2}

(2 (5 cos (x) - 4))^{2} = 2^{2} (5 cos (x) - 4)^{2}

(2 (5 cos (x) - 4))^{2}

Aplicar las leyes de los exponentes:

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

= 2^{2} (5 cos (x) - 4)^{2}

= 2^{2} (5 cos (x) - 4)^{2} (2 (5 sin (x) - 3))^{2}

(2 (5 sin (x) - 3))^{2} = 2^{2} (5 sin (x) - 3)^{2}

(2 (5 sin (x) - 3))^{2}

Aplicar las leyes de los exponentes:

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

= 2^{2} (5 sin (x) - 3)^{2}

= 2^{2} \cdot 2^{2} (5 cos (x) - 4)^{2} (5 sin (x) - 3)^{2}

Simplificar

2^{2} (5 cos (x) - 4)^{2} \cdot 2^{2} (5 sin (x) - 3)^{2} : 2^{4} (5 cos (x) - 4)^{2} (5 sin (x) - 3)^{2}

2^{2} (5 cos (x) - 4)^{2} \cdot 2^{2} (5 sin (x) - 3)^{2}

Aplicar las leyes de los exponentes:

a^{b} \cdot a^{c} = a^{b + c}

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

= (5 cos (x) - 4)^{2} \cdot 2^{2 + 2} (5 sin (x) - 3)^{2}

Sumar:

2 + 2 = 4

= (5 cos (x) - 4)^{2} \cdot 2^{4} (5 sin (x) - 3)^{2}

= 2^{4} (5 cos (x) - 4)^{2} (5 sin (x) - 3)^{2}

Aplicar la siguiente propiedad de los radicales:

asumiendo que

a \geq 0, b \geq 0

= 2^{4} (5 cos (x) - 4)^{2} (5 sin (x) - 3)^{2}

Aplicar las leyes de los exponentes:

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

2^{4} = 2^{2 \cdot 2} = (2^{2})^{2}

= (5 cos (x) - 4)^{2} (2^{2})^{2} (5 sin (x) - 3)^{2}

Aplicar la siguiente propiedad de los radicales:

asumiendo que

a \geq 0

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

= 2^{2} (5 cos (x) - 4)^{2} (5 sin (x) - 3)^{2}

2^{2} = 4

= 4 (5 cos (x) - 4)^{2} (5 sin (x) - 3)^{2}

4 (5 cos (x) - 4)^{2} (5 sin (x) - 3)^{2} = 0

Resolver cada parte por separado

(5 cos (x) - 4)^{2} = 0 or (5 sin (x) - 3)^{2} = 0

(5 cos (x) - 4)^{2} = 0 : x = arccos (\frac{4}{5}) + 2 πn, x = 2 π - arccos (\frac{4}{5}) + 2 πn

(5 cos (x) - 4)^{2} = 0

Elevar al cuadrado ambos lados:

(5 cos (x) - 4)^{2} = 0

(5 cos (x) - 4)^{2} = 0

((5 cos (x) - 4)^{2})^{2} = 0^{2}

Desarrollar

((5 cos (x) - 4)^{2})^{2} : (5 cos (x) - 4)^{2}

((5 cos (x) - 4)^{2})^{2}

Aplicar las leyes de los exponentes:

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

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

Aplicar las leyes de los exponentes:

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

= ((5 cos (x) - 4)^{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

= (5 cos (x) - 4)^{2}

Desarrollar

0^{2} : 0

0^{2}

Aplicar la regla

0^{a} = 0

= 0

(5 cos (x) - 4)^{2} = 0

(5 cos (x) - 4)^{2} = 0

Resolver

(5 cos (x) - 4)^{2} = 0 : cos (x) = \frac{4}{5}

(5 cos (x) - 4)^{2} = 0

Utilizando el teorema de factor cero: si

ab = 0

entonces

a = 0

b = 0

Resolver

5 cos (x) - 4 = 0 : cos (x) = \frac{4}{5}

5 cos (x) - 4 = 0

Mover

4

al lado derecho

5 cos (x) - 4 = 0

Sumar

4

a ambos lados

5 cos (x) - 4 + 4 = 0 + 4

Simplificar

5 cos (x) = 4

5 cos (x) = 4

Dividir ambos lados entre

5

5 cos (x) = 4

Dividir ambos lados entre

5

\frac{5 cos ( x )}{5} = \frac{4}{5}

Simplificar

cos (x) = \frac{4}{5}

cos (x) = \frac{4}{5}

La solución a la ecuación de segundo grado es:

cos (x) = \frac{4}{5}

cos (x) = \frac{4}{5}

Verificar las soluciones:

cos (x) = \frac{4}{5}

Verdadero

Verificar las soluciones sustituyéndolas en

(5 cos (x) - 4)^{2} = 0

Quitar las que no concuerden con la ecuación.

Sustituir

cos (x) = \frac{4}{5} :

Verdadero

(5 (\frac{4}{5}) - 4)^{2} = 0

(5 (\frac{4}{5}) - 4)^{2} = 0

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

Quitar los parentesis:

(a) = a

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

(5 \cdot \frac{4}{5} - 4)^{2} = 0

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

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

5 \cdot \frac{4}{5}

Multiplicar fracciones:

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

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

Eliminar los terminos comunes:

5

= 4

= (4 - 4)^{2}

Restar:

4 - 4 = 0

= 0^{2}

Aplicar la regla

0^{a} = 0

= 0

= 0

Aplicar la regla

0 = 0

= 0

0 = 0

Verdadero

La solución es

cos (x) = \frac{4}{5}

Aplicar propiedades trigonométricas inversas

cos (x) = \frac{4}{5}

Soluciones generales para

cos (x) = \frac{4}{5}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = 2 π - arccos (a) + 2 πn

x = arccos (\frac{4}{5}) + 2 πn, x = 2 π - arccos (\frac{4}{5}) + 2 πn

x = arccos (\frac{4}{5}) + 2 πn, x = 2 π - arccos (\frac{4}{5}) + 2 πn

(5 sin (x) - 3)^{2} = 0 : x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

(5 sin (x) - 3)^{2} = 0

Elevar al cuadrado ambos lados:

(5 sin (x) - 3)^{2} = 0

(5 sin (x) - 3)^{2} = 0

((5 sin (x) - 3)^{2})^{2} = 0^{2}

Desarrollar

((5 sin (x) - 3)^{2})^{2} : (5 sin (x) - 3)^{2}

((5 sin (x) - 3)^{2})^{2}

Aplicar las leyes de los exponentes:

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

= (((5 sin (x) - 3)^{2})^{\frac{1}{2}})^{2}

Aplicar las leyes de los exponentes:

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

= ((5 sin (x) - 3)^{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

= (5 sin (x) - 3)^{2}

Desarrollar

0^{2} : 0

0^{2}

Aplicar la regla

0^{a} = 0

= 0

(5 sin (x) - 3)^{2} = 0

(5 sin (x) - 3)^{2} = 0

Resolver

(5 sin (x) - 3)^{2} = 0 : sin (x) = \frac{3}{5}

(5 sin (x) - 3)^{2} = 0

Utilizando el teorema de factor cero: si

ab = 0

entonces

a = 0

b = 0

Resolver

5 sin (x) - 3 = 0 : sin (x) = \frac{3}{5}

5 sin (x) - 3 = 0

Mover

3

al lado derecho

5 sin (x) - 3 = 0

Sumar

3

a ambos lados

5 sin (x) - 3 + 3 = 0 + 3

Simplificar

5 sin (x) = 3

5 sin (x) = 3

Dividir ambos lados entre

5

5 sin (x) = 3

Dividir ambos lados entre

5

\frac{5 sin ( x )}{5} = \frac{3}{5}

Simplificar

sin (x) = \frac{3}{5}

sin (x) = \frac{3}{5}

La solución a la ecuación de segundo grado es:

sin (x) = \frac{3}{5}

sin (x) = \frac{3}{5}

Verificar las soluciones:

sin (x) = \frac{3}{5}

Verdadero

Verificar las soluciones sustituyéndolas en

(5 sin (x) - 3)^{2} = 0

Quitar las que no concuerden con la ecuación.

Sustituir

sin (x) = \frac{3}{5} :

Verdadero

(5 (\frac{3}{5}) - 3)^{2} = 0

(5 (\frac{3}{5}) - 3)^{2} = 0

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

Quitar los parentesis:

(a) = a

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

(5 \cdot \frac{3}{5} - 3)^{2} = 0

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

5 \cdot \frac{3}{5} = 3

5 \cdot \frac{3}{5}

Multiplicar fracciones:

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

= \frac{3 \cdot 5}{5}

Eliminar los terminos comunes:

5

= 3

= (3 - 3)^{2}

Restar:

3 - 3 = 0

= 0^{2}

Aplicar la regla

0^{a} = 0

= 0

= 0

Aplicar la regla

0 = 0

= 0

0 = 0

Verdadero

La solución es

sin (x) = \frac{3}{5}

Aplicar propiedades trigonométricas inversas

sin (x) = \frac{3}{5}

Soluciones generales para

sin (x) = \frac{3}{5}

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

x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

Combinar toda las soluciones

x = arccos (\frac{4}{5}) + 2 πn, x = 2 π - arccos (\frac{4}{5}) + 2 πn, x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

Mezclar intervalos sobrepuestos

x = 2 π - arccos (\frac{4}{5}) + 2 πn, x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

Mostrar soluciones en forma decimal

x = 2 π - 0.64350 \dots + 2 πn, x = 0.64350 \dots + 2 πn, x = π - 0.64350 \dots + 2 πn

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3 (1 + cos (θ)) = 3

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