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8sin^2(x)+6cos^2(x)=10

Solución

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

Solución

Sin soluci \overset{o}{ˊ} n para x \in R

Pasos de solución

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

Restar

10

de ambos lados

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

= - 10 + 6 (1 - sin^{2} (x)) + 8 sin^{2} (x)

Simplificar

- 10 + 6 (1 - sin^{2} (x)) + 8 sin^{2} (x) : 2 sin^{2} (x) - 4

- 10 + 6 (1 - sin^{2} (x)) + 8 sin^{2} (x)

Expandir

6 (1 - sin^{2} (x)) : 6 - 6 sin^{2} (x)

6 (1 - sin^{2} (x))

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = 6, b = 1, c = sin^{2} (x)

= 6 \cdot 1 - 6 sin^{2} (x)

Multiplicar los numeros:

6 \cdot 1 = 6

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

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

Simplificar

- 10 + 6 - 6 sin^{2} (x) + 8 sin^{2} (x) : 2 sin^{2} (x) - 4

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

Sumar elementos similares:

- 6 sin^{2} (x) + 8 sin^{2} (x) = 2 sin^{2} (x)

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

Sumar/restar lo siguiente:

- 10 + 6 = - 4

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

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

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

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

Usando el método de sustitución

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

Sea:

sin (x) = u

- 4 + 2 u^{2} = 0

- 4 + 2 u^{2} = 0 : u = 2, u = - 2

- 4 + 2 u^{2} = 0

Mover

4

al lado derecho

- 4 + 2 u^{2} = 0

Sumar

4

a ambos lados

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

Simplificar

2 u^{2} = 4

2 u^{2} = 4

Dividir ambos lados entre

2

2 u^{2} = 4

Dividir ambos lados entre

2

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

Simplificar

u^{2} = 2

u^{2} = 2

Para

x^{2} = f (a)

las soluciones son

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

u = 2, u = - 2

Sustituir en la ecuación

u = sin (x)

sin (x) = 2, sin (x) = - 2

sin (x) = 2, sin (x) = - 2

sin (x) = 2 :

Sin solución

sin (x) = 2

- 1 \leq sin (x) \leq 1

Sin soluci \overset{o}{ˊ} n

sin (x) = - 2 :

Sin solución

sin (x) = - 2

- 1 \leq sin (x) \leq 1

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

Sin soluci \overset{o}{ˊ} n para x \in R

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