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Popular >

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

Solución

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

Solución

x = 0.95531 \dots + 2 πn, x = π - 0.95531 \dots + 2 πn, x = - 0.95531 \dots + 2 πn, x = π + 0.95531 \dots + 2 πn

Grados

x = 54.73561 \dots^{\circ} + 36 0^{\circ} n, x = 125.26438 \dots^{\circ} + 36 0^{\circ} n, x = - 54.73561 \dots^{\circ} + 36 0^{\circ} n, x = 234.73561 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

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

Restar

3

de ambos lados

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad trigonométrica del ángulo doble:

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

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

Simplificar

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

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

Sumar elementos similares:

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

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

Sumar/restar lo siguiente:

- 3 + 1 = - 2

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

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

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

Usando el método de sustitución

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

Sea:

sin (x) = u

- 2 + 3 u^{2} = 0

- 2 + 3 u^{2} = 0 : u = \frac{2}{3}, u = - \frac{2}{3}

- 2 + 3 u^{2} = 0

Mover

2

al lado derecho

- 2 + 3 u^{2} = 0

Sumar

2

a ambos lados

- 2 + 3 u^{2} + 2 = 0 + 2

Simplificar

3 u^{2} = 2

3 u^{2} = 2

Dividir ambos lados entre

3

3 u^{2} = 2

Dividir ambos lados entre

3

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

Simplificar

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

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

Para

x^{2} = f (a)

las soluciones son

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

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

Sustituir en la ecuación

u = sin (x)

sin (x) = \frac{2}{3}, sin (x) = - \frac{2}{3}

sin (x) = \frac{2}{3}, sin (x) = - \frac{2}{3}

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

sin (x) = \frac{2}{3}

Aplicar propiedades trigonométricas inversas

sin (x) = \frac{2}{3}

Soluciones generales para

sin (x) = \frac{2}{3}

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

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

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

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

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

Aplicar propiedades trigonométricas inversas

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

Soluciones generales para

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

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

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

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

Combinar toda las soluciones

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

Mostrar soluciones en forma decimal

x = 0.95531 \dots + 2 πn, x = π - 0.95531 \dots + 2 πn, x = - 0.95531 \dots + 2 πn, x = π + 0.95531 \dots + 2 πn

Ejemplos populares

sin(θ)=-3/8

sin (θ) = - \frac{3}{8}

3sec(x)+5-cos(x)=cos(x)

3 sec (x) + 5 - cos (x) = cos (x)

2sin(x)-cos(x)sin(x)=0

2 sin (x) - cos (x) sin (x) = 0

sin(2x-20)=-cos(3x+50)

sin (2 x - 20) = - cos (3 x + 50)

(sin(60))/(174.36)=(sin(x))/(200)

\frac{sin ( 6 0 ^{\circ} )}{174.36} = \frac{sin ( x )}{200}