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2sec^2(x)-13sec(x)+20=0,2pi<= x<= 4pi

Solución

2 sec^{2} (x) - 13 sec (x) + 20 = 0, 2 π \leq x \leq 4 π

Solución

x = 1.31811 \dots + 2 π, x = 4 π - 1.31811 \dots, x = 1.15927 \dots + 2 π, x = 4 π - 1.15927 \dots

Grados

x = 435.52248 \dots^{\circ}, x = 644.47751 \dots^{\circ}, x = 426.42182 \dots^{\circ}, x = 653.57817 \dots^{\circ}

Pasos de solución

2 sec^{2} (x) - 13 sec (x) + 20 = 0, 2 π \leq x \leq 4 π

Usando el método de sustitución

2 sec^{2} (x) - 13 sec (x) + 20 = 0

Sea:

sec (x) = u

2 u^{2} - 13 u + 20 = 0

2 u^{2} - 13 u + 20 = 0 : u = 4, u = \frac{5}{2}

2 u^{2} - 13 u + 20 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

2 u^{2} - 13 u + 20 = 0

Formula general para ecuaciones de segundo grado:

Para

a = 2, b = - 13, c = 20

u_{1, 2} = \frac{- ( - 13 ) \pm ( - 13 ) ^{2} - 4 \cdot 2 \cdot 20}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 13 ) \pm ( - 13 ) ^{2} - 4 \cdot 2 \cdot 20}{2 \cdot 2}

(- 13)^{2} - 4 \cdot 2 \cdot 20 = 3

(- 13)^{2} - 4 \cdot 2 \cdot 20

Aplicar las leyes de los exponentes:

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

n

es par

(- 13)^{2} = 1 3^{2}

= 1 3^{2} - 4 \cdot 2 \cdot 20

Multiplicar los numeros:

4 \cdot 2 \cdot 20 = 160

= 1 3^{2} - 160

1 3^{2} = 169

= 169 - 160

Restar:

169 - 160 = 9

= 9

Descomponer el número en factores primos:

9 = 3^{2}

= 3^{2}

Aplicar las leyes de los exponentes:

3^{2} = 3

= 3

u_{1, 2} = \frac{- ( - 13 ) \pm 3}{2 \cdot 2}

Separar las soluciones

u_{1} = \frac{- ( - 13 ) + 3}{2 \cdot 2}, u_{2} = \frac{- ( - 13 ) - 3}{2 \cdot 2}

u = \frac{- ( - 13 ) + 3}{2 \cdot 2} : 4

\frac{- ( - 13 ) + 3}{2 \cdot 2}

Aplicar la regla

- (- a) = a

= \frac{13 + 3}{2 \cdot 2}

Sumar:

13 + 3 = 16

= \frac{16}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{16}{4}

Dividir:

\frac{16}{4} = 4

= 4

u = \frac{- ( - 13 ) - 3}{2 \cdot 2} : \frac{5}{2}

\frac{- ( - 13 ) - 3}{2 \cdot 2}

Aplicar la regla

- (- a) = a

= \frac{13 - 3}{2 \cdot 2}

Restar:

13 - 3 = 10

= \frac{10}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{10}{4}

Eliminar los terminos comunes:

2

= \frac{5}{2}

Las soluciones a la ecuación de segundo grado son:

u = 4, u = \frac{5}{2}

Sustituir en la ecuación

u = sec (x)

sec (x) = 4, sec (x) = \frac{5}{2}

sec (x) = 4, sec (x) = \frac{5}{2}

sec (x) = 4, 2 π \leq x \leq 4 π : x = arcsec (4) + 2 π, x = 4 π - arcsec (4)

sec (x) = 4, 2 π \leq x \leq 4 π

Aplicar propiedades trigonométricas inversas

sec (x) = 4

Soluciones generales para

sec (x) = 4

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = 2 π - arcsec (a) + 2 πn

x = arcsec (4) + 2 πn, x = 2 π - arcsec (4) + 2 πn

x = arcsec (4) + 2 πn, x = 2 π - arcsec (4) + 2 πn

Soluciones para el rango

2 π \leq x \leq 4 π

x = arcsec (4) + 2 π, x = 4 π - arcsec (4)

sec (x) = \frac{5}{2}, 2 π \leq x \leq 4 π : x = arcsec (\frac{5}{2}) + 2 π, x = 4 π - arcsec (\frac{5}{2})

sec (x) = \frac{5}{2}, 2 π \leq x \leq 4 π

Aplicar propiedades trigonométricas inversas

sec (x) = \frac{5}{2}

Soluciones generales para

sec (x) = \frac{5}{2}

sec (x) = a \Rightarrow x = arcsec (a) + 2 πn, x = 2 π - arcsec (a) + 2 πn

x = arcsec (\frac{5}{2}) + 2 πn, x = 2 π - arcsec (\frac{5}{2}) + 2 πn

x = arcsec (\frac{5}{2}) + 2 πn, x = 2 π - arcsec (\frac{5}{2}) + 2 πn

Soluciones para el rango

2 π \leq x \leq 4 π

x = arcsec (\frac{5}{2}) + 2 π, x = 4 π - arcsec (\frac{5}{2})

Combinar toda las soluciones

x = arcsec (4) + 2 π, x = 4 π - arcsec (4), x = arcsec (\frac{5}{2}) + 2 π, x = 4 π - arcsec (\frac{5}{2})

Mostrar soluciones en forma decimal

x = 1.31811 \dots + 2 π, x = 4 π - 1.31811 \dots, x = 1.15927 \dots + 2 π, x = 4 π - 1.15927 \dots

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20sin(x)=20cos(x)

20 sin (x) = 20 cos (x)

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