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4sin^2(x-60)=3

Solución

4 sin^{2} (x - 6 0^{\circ}) = 3

Solución

x = 12 0^{\circ} + 36 0^{\circ} n, x = 36 0^{\circ} n + 18 0^{\circ}, x = 36 0^{\circ} n + 30 0^{\circ}, x = 36 0^{\circ} n + 36 0^{\circ}

radianes

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

Pasos de solución

4 sin^{2} (x - 6 0^{\circ}) = 3

Usando el método de sustitución

4 sin^{2} (x - 6 0^{\circ}) = 3

Sea:

sin (x - 6 0^{\circ}) = u

4 u^{2} = 3

4 u^{2} = 3 : u = \frac{3}{2}, u = - \frac{3}{2}

4 u^{2} = 3

Dividir ambos lados entre

4

4 u^{2} = 3

Dividir ambos lados entre

4

\frac{4 u ^{2}}{4} = \frac{3}{4}

Simplificar

u^{2} = \frac{3}{4}

u^{2} = \frac{3}{4}

Para

x^{2} = f (a)

las soluciones son

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

u = \frac{3}{4}, u = - \frac{3}{4}

\frac{3}{4} = \frac{3}{2}

\frac{3}{4}

Aplicar la siguiente propiedad de los radicales:

asumiendo que

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Descomponer el número en factores primos:

4 = 2^{2}

= 2^{2}

Aplicar las leyes de los exponentes:

2^{2} = 2

= 2

= \frac{3}{2}

- \frac{3}{4} = - \frac{3}{2}

- \frac{3}{4}

Simplificar

\frac{3}{4} : \frac{3}{2}

\frac{3}{4}

Aplicar la siguiente propiedad de los radicales:

asumiendo que

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

Descomponer el número en factores primos:

4 = 2^{2}

= 2^{2}

Aplicar las leyes de los exponentes:

2^{2} = 2

= 2

= \frac{3}{2}

= - \frac{3}{2}

u = \frac{3}{2}, u = - \frac{3}{2}

Sustituir en la ecuación

u = sin (x - 6 0^{\circ})

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

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

sin (x - 6 0^{\circ}) = \frac{3}{2} : x = 12 0^{\circ} + 36 0^{\circ} n, x = 36 0^{\circ} n + 18 0^{\circ}

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

Soluciones generales para

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

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}

x - 6 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n, x - 6 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n

x - 6 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n, x - 6 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n

Resolver

x - 6 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n : x = 12 0^{\circ} + 36 0^{\circ} n

x - 6 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n

Mover

6 0^{\circ}

al lado derecho

x - 6 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n

Sumar

6 0^{\circ}

a ambos lados

x - 6 0^{\circ} + 6 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Simplificar

x - 6 0^{\circ} + 6 0^{\circ} = 6 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Simplificar

x - 6 0^{\circ} + 6 0^{\circ} : x

x - 6 0^{\circ} + 6 0^{\circ}

Sumar elementos similares:

- 6 0^{\circ} + 6 0^{\circ} = 0

= x

Simplificar

6 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ} : 12 0^{\circ} + 36 0^{\circ} n

6 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Agrupar términos semejantes

= 6 0^{\circ} + 6 0^{\circ} + 36 0^{\circ} n

Combinar las fracciones usando el mínimo común denominador:

12 0^{\circ}

Aplicar la regla

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

= \frac{18 0 ^{\circ} + 18 0 ^{\circ}}{3}

Sumar elementos similares:

18 0^{\circ} + 18 0^{\circ} = 36 0^{\circ}

= 12 0^{\circ}

= 12 0^{\circ} + 36 0^{\circ} n

x = 12 0^{\circ} + 36 0^{\circ} n

x = 12 0^{\circ} + 36 0^{\circ} n

x = 12 0^{\circ} + 36 0^{\circ} n

Resolver

x - 6 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 18 0^{\circ}

x - 6 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n

Mover

6 0^{\circ}

al lado derecho

x - 6 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n

Sumar

6 0^{\circ}

a ambos lados

x - 6 0^{\circ} + 6 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Simplificar

x - 6 0^{\circ} + 6 0^{\circ} = 12 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Simplificar

x - 6 0^{\circ} + 6 0^{\circ} : x

x - 6 0^{\circ} + 6 0^{\circ}

Sumar elementos similares:

- 6 0^{\circ} + 6 0^{\circ} = 0

= x

Simplificar

12 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ} : 36 0^{\circ} n + 18 0^{\circ}

12 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Agrupar términos semejantes

= 36 0^{\circ} n + 6 0^{\circ} + 12 0^{\circ}

Combinar las fracciones usando el mínimo común denominador:

18 0^{\circ}

Aplicar la regla

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

= \frac{18 0 ^{\circ} + 36 0 ^{\circ}}{3}

Sumar elementos similares:

18 0^{\circ} + 36 0^{\circ} = 54 0^{\circ}

= 18 0^{\circ}

Dividir:

\frac{3}{3} = 1

= 18 0^{\circ}

= 36 0^{\circ} n + 18 0^{\circ}

x = 36 0^{\circ} n + 18 0^{\circ}

x = 36 0^{\circ} n + 18 0^{\circ}

x = 36 0^{\circ} n + 18 0^{\circ}

x = 12 0^{\circ} + 36 0^{\circ} n, x = 36 0^{\circ} n + 18 0^{\circ}

sin (x - 6 0^{\circ}) = - \frac{3}{2} : x = 36 0^{\circ} n + 30 0^{\circ}, x = 36 0^{\circ} n + 36 0^{\circ}

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

Soluciones generales para

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

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}

x - 6 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n, x - 6 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

x - 6 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n, x - 6 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

Resolver

x - 6 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 30 0^{\circ}

x - 6 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n

Mover

6 0^{\circ}

al lado derecho

x - 6 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n

Sumar

6 0^{\circ}

a ambos lados

x - 6 0^{\circ} + 6 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Simplificar

x - 6 0^{\circ} + 6 0^{\circ} = 24 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Simplificar

x - 6 0^{\circ} + 6 0^{\circ} : x

x - 6 0^{\circ} + 6 0^{\circ}

Sumar elementos similares:

- 6 0^{\circ} + 6 0^{\circ} = 0

= x

Simplificar

24 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ} : 36 0^{\circ} n + 30 0^{\circ}

24 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Agrupar términos semejantes

= 36 0^{\circ} n + 6 0^{\circ} + 24 0^{\circ}

Combinar las fracciones usando el mínimo común denominador:

30 0^{\circ}

Aplicar la regla

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

= \frac{18 0 ^{\circ} + 72 0 ^{\circ}}{3}

Sumar elementos similares:

18 0^{\circ} + 72 0^{\circ} = 90 0^{\circ}

= 30 0^{\circ}

= 36 0^{\circ} n + 30 0^{\circ}

x = 36 0^{\circ} n + 30 0^{\circ}

x = 36 0^{\circ} n + 30 0^{\circ}

x = 36 0^{\circ} n + 30 0^{\circ}

Resolver

x - 6 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n : x = 36 0^{\circ} n + 36 0^{\circ}

x - 6 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

Mover

6 0^{\circ}

al lado derecho

x - 6 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n

Sumar

6 0^{\circ}

a ambos lados

x - 6 0^{\circ} + 6 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Simplificar

x - 6 0^{\circ} + 6 0^{\circ} = 30 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Simplificar

x - 6 0^{\circ} + 6 0^{\circ} : x

x - 6 0^{\circ} + 6 0^{\circ}

Sumar elementos similares:

- 6 0^{\circ} + 6 0^{\circ} = 0

= x

Simplificar

30 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ} : 36 0^{\circ} n + 36 0^{\circ}

30 0^{\circ} + 36 0^{\circ} n + 6 0^{\circ}

Agrupar términos semejantes

= 36 0^{\circ} n + 6 0^{\circ} + 30 0^{\circ}

Combinar las fracciones usando el mínimo común denominador:

36 0^{\circ}

Aplicar la regla

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

= \frac{18 0 ^{\circ} + 90 0 ^{\circ}}{3}

Sumar elementos similares:

18 0^{\circ} + 90 0^{\circ} = 108 0^{\circ}

= 36 0^{\circ}

Dividir:

\frac{6}{3} = 2

= 36 0^{\circ}

= 36 0^{\circ} n + 36 0^{\circ}

x = 36 0^{\circ} n + 36 0^{\circ}

x = 36 0^{\circ} n + 36 0^{\circ}

x = 36 0^{\circ} n + 36 0^{\circ}

x = 36 0^{\circ} n + 30 0^{\circ}, x = 36 0^{\circ} n + 36 0^{\circ}

Combinar toda las soluciones

x = 12 0^{\circ} + 36 0^{\circ} n, x = 36 0^{\circ} n + 18 0^{\circ}, x = 36 0^{\circ} n + 30 0^{\circ}, x = 36 0^{\circ} n + 36 0^{\circ}

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