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sin^2(x)+5cos^2(x)=3

Solución

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

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

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

Restar

3

de ambos lados

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

- 3 + sin^{2} (x) + 5 (1 - sin^{2} (x)) : - 4 sin^{2} (x) + 2

- 3 + sin^{2} (x) + 5 (1 - sin^{2} (x))

Expandir

5 (1 - sin^{2} (x)) : 5 - 5 sin^{2} (x)

5 (1 - sin^{2} (x))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 5, b = 1, c = sin^{2} (x)

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

Multiplicar los numeros:

5 \cdot 1 = 5

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

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

Simplificar

- 3 + sin^{2} (x) + 5 - 5 sin^{2} (x) : - 4 sin^{2} (x) + 2

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

Agrupar términos semejantes

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

Sumar elementos similares:

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

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

Sumar/restar lo siguiente:

- 3 + 5 = 2

= - 4 sin^{2} (x) + 2

= - 4 sin^{2} (x) + 2

= - 4 sin^{2} (x) + 2

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

Usando el método de sustitución

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

Sea:

sin (x) = u

2 - 4 u^{2} = 0

2 - 4 u^{2} = 0 : u = \frac{1}{2}, u = - \frac{1}{2}

2 - 4 u^{2} = 0

Mover

2

al lado derecho

2 - 4 u^{2} = 0

Restar

2

de ambos lados

2 - 4 u^{2} - 2 = 0 - 2

Simplificar

- 4 u^{2} = - 2

- 4 u^{2} = - 2

Dividir ambos lados entre

- 4

- 4 u^{2} = - 2

Dividir ambos lados entre

- 4

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

Simplificar

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

Simplificar

\frac{- 4 u ^{2}}{- 4} : u^{2}

\frac{- 4 u ^{2}}{- 4}

Aplicar las propiedades de las fracciones:

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

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

Dividir:

\frac{4}{4} = 1

= u^{2}

Simplificar

\frac{- 2}{- 4} : \frac{1}{2}

\frac{- 2}{- 4}

Aplicar las propiedades de las fracciones:

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

= \frac{2}{4}

Eliminar los terminos comunes:

2

= \frac{1}{2}

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

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}

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

sin(x-pi/2)+cos(x+pi/2)=0

sin (x - \frac{π}{2}) + cos (x + \frac{π}{2}) = 0

(sqrt(3))/2 =sin(arcsin(0)+c)

\frac{3}{2} = sin (arcsin (0) + c)

1+2sin^2(x)-3sin(x)=0

1 + 2 sin^{2} (x) - 3 sin (x) = 0

4tan^2(x)=-sqrt(3)tan(x)+tan^2(x)

4 tan^{2} (x) = - 3 tan (x) + tan^{2} (x)

1/2 = 1/2*cos(4.2t+pi/6)

\frac{1}{2} = \frac{1}{2} \cdot cos (4.2 t + \frac{π}{6})