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Popular Trigonometría >

sin^2(2θ)-5sin(2θ)-1=0

Solución

sin^{2} (2 θ) - 5 sin (2 θ) - 1 = 0

Solución

θ = \frac{- 0.19379 \dots}{2} + πn, θ = \frac{π}{2} + \frac{0.19379 \dots}{2} + πn

Grados

θ = - 5.55176 \dots^{\circ} + 18 0^{\circ} n, θ = 95.55176 \dots^{\circ} + 18 0^{\circ} n

Pasos de solución

sin^{2} (2 θ) - 5 sin (2 θ) - 1 = 0

Usando el método de sustitución

sin^{2} (2 θ) - 5 sin (2 θ) - 1 = 0

Sea:

sin (2 θ) = u

u^{2} - 5 u - 1 = 0

u^{2} - 5 u - 1 = 0 : u = \frac{5 + 29}{2}, u = \frac{5 - 29}{2}

u^{2} - 5 u - 1 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

u^{2} - 5 u - 1 = 0

Formula general para ecuaciones de segundo grado:

Para

a = 1, b = - 5, c = - 1

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

u_{1, 2} = \frac{- ( - 5 ) \pm ( - 5 ) ^{2} - 4 \cdot 1 \cdot ( - 1 )}{2 \cdot 1}

(- 5)^{2} - 4 \cdot 1 \cdot (- 1) = 29

(- 5)^{2} - 4 \cdot 1 \cdot (- 1)

Aplicar la regla

- (- a) = a

= (- 5)^{2} + 4 \cdot 1 \cdot 1

Aplicar las leyes de los exponentes:

(- a)^{n} = a^{n},

n

es par

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

= 5^{2} + 4 \cdot 1 \cdot 1

Multiplicar los numeros:

4 \cdot 1 \cdot 1 = 4

= 5^{2} + 4

5^{2} = 25

= 25 + 4

Sumar:

25 + 4 = 29

= 29

u_{1, 2} = \frac{- ( - 5 ) \pm 29}{2 \cdot 1}

Separar las soluciones

u_{1} = \frac{- ( - 5 ) + 29}{2 \cdot 1}, u_{2} = \frac{- ( - 5 ) - 29}{2 \cdot 1}

u = \frac{- ( - 5 ) + 29}{2 \cdot 1} : \frac{5 + 29}{2}

\frac{- ( - 5 ) + 29}{2 \cdot 1}

Aplicar la regla

- (- a) = a

= \frac{5 + 29}{2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{5 + 29}{2}

u = \frac{- ( - 5 ) - 29}{2 \cdot 1} : \frac{5 - 29}{2}

\frac{- ( - 5 ) - 29}{2 \cdot 1}

Aplicar la regla

- (- a) = a

= \frac{5 - 29}{2 \cdot 1}

Multiplicar los numeros:

2 \cdot 1 = 2

= \frac{5 - 29}{2}

Las soluciones a la ecuación de segundo grado son:

u = \frac{5 + 29}{2}, u = \frac{5 - 29}{2}

Sustituir en la ecuación

u = sin (2 θ)

sin (2 θ) = \frac{5 + 29}{2}, sin (2 θ) = \frac{5 - 29}{2}

sin (2 θ) = \frac{5 + 29}{2}, sin (2 θ) = \frac{5 - 29}{2}

sin (2 θ) = \frac{5 + 29}{2} :

Sin solución

sin (2 θ) = \frac{5 + 29}{2}

- 1 \leq sin (x) \leq 1

Sin soluci \overset{o}{ˊ} n

sin (2 θ) = \frac{5 - 29}{2} : θ = \frac{arcsin ( \frac{5 - 29}{2} )}{2} + πn, θ = \frac{π}{2} + \frac{arcsin ( \frac{29 - 5}{2} )}{2} + πn

sin (2 θ) = \frac{5 - 29}{2}

Aplicar propiedades trigonométricas inversas

sin (2 θ) = \frac{5 - 29}{2}

Soluciones generales para

sin (2 θ) = \frac{5 - 29}{2}

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

2 θ = arcsin (\frac{5 - 29}{2}) + 2 πn, 2 θ = π + arcsin (- \frac{5 - 29}{2}) + 2 πn

2 θ = arcsin (\frac{5 - 29}{2}) + 2 πn, 2 θ = π + arcsin (- \frac{5 - 29}{2}) + 2 πn

Resolver

2 θ = arcsin (\frac{5 - 29}{2}) + 2 πn : θ = \frac{arcsin ( \frac{5 - 29}{2} )}{2} + πn

2 θ = arcsin (\frac{5 - 29}{2}) + 2 πn

Dividir ambos lados entre

2

2 θ = arcsin (\frac{5 - 29}{2}) + 2 πn

Dividir ambos lados entre

2

\frac{2 θ}{2} = \frac{arcsin ( \frac{5 - 29}{2} )}{2} + \frac{2 πn}{2}

Simplificar

θ = \frac{arcsin ( \frac{5 - 29}{2} )}{2} + πn

θ = \frac{arcsin ( \frac{5 - 29}{2} )}{2} + πn

Resolver

2 θ = π + arcsin (- \frac{5 - 29}{2}) + 2 πn : θ = \frac{π}{2} + \frac{arcsin ( \frac{29 - 5}{2} )}{2} + πn

2 θ = π + arcsin (- \frac{5 - 29}{2}) + 2 πn

- \frac{5 - 29}{2} = - \frac{- ( 29 - 5 )}{2} = \frac{29 - 5}{2}

2 θ = π + arcsin (\frac{29 - 5}{2}) + 2 πn

Dividir ambos lados entre

2

2 θ = π + arcsin (\frac{29 - 5}{2}) + 2 πn

Dividir ambos lados entre

2

\frac{2 θ}{2} = \frac{π}{2} + \frac{arcsin ( \frac{29 - 5}{2} )}{2} + \frac{2 πn}{2}

Simplificar

θ = \frac{π}{2} + \frac{arcsin ( \frac{29 - 5}{2} )}{2} + πn

θ = \frac{π}{2} + \frac{arcsin ( \frac{29 - 5}{2} )}{2} + πn

θ = \frac{arcsin ( \frac{5 - 29}{2} )}{2} + πn, θ = \frac{π}{2} + \frac{arcsin ( \frac{29 - 5}{2} )}{2} + πn

Combinar toda las soluciones

θ = \frac{arcsin ( \frac{5 - 29}{2} )}{2} + πn, θ = \frac{π}{2} + \frac{arcsin ( \frac{29 - 5}{2} )}{2} + πn

Mostrar soluciones en forma decimal

θ = \frac{- 0.19379 \dots}{2} + πn, θ = \frac{π}{2} + \frac{0.19379 \dots}{2} + πn

Gráfica

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