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Popular Trigonometría >

2tan^2(x)-7sec(x)+5=0

Solución

2 tan^{2} (x) - 7 sec (x) + 5 = 0

Solución

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

Grados

x = 70.52877 \dots^{\circ} + 36 0^{\circ} n, x = 289.47122 \dots^{\circ} + 36 0^{\circ} n

Pasos de solución

2 tan^{2} (x) - 7 sec (x) + 5 = 0

Re-escribir usando identidades trigonométricas

5 + 2 tan^{2} (x) - 7 sec (x)

Utilizar la identidad pitagórica:

tan^{2} (x) + 1 = sec^{2} (x)

tan^{2} (x) = sec^{2} (x) - 1

= 5 + 2 (sec^{2} (x) - 1) - 7 sec (x)

Simplificar

5 + 2 (sec^{2} (x) - 1) - 7 sec (x) : 2 sec^{2} (x) - 7 sec (x) + 3

5 + 2 (sec^{2} (x) - 1) - 7 sec (x)

Expandir

2 (sec^{2} (x) - 1) : 2 sec^{2} (x) - 2

2 (sec^{2} (x) - 1)

Poner los parentesis utilizando:

a (b - c) = ab - a c

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

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

Multiplicar los numeros:

2 \cdot 1 = 2

= 2 sec^{2} (x) - 2

= 5 + 2 sec^{2} (x) - 2 - 7 sec (x)

Simplificar

5 + 2 sec^{2} (x) - 2 - 7 sec (x) : 2 sec^{2} (x) - 7 sec (x) + 3

5 + 2 sec^{2} (x) - 2 - 7 sec (x)

Agrupar términos semejantes

= 2 sec^{2} (x) - 7 sec (x) + 5 - 2

Sumar/restar lo siguiente:

5 - 2 = 3

= 2 sec^{2} (x) - 7 sec (x) + 3

= 2 sec^{2} (x) - 7 sec (x) + 3

= 2 sec^{2} (x) - 7 sec (x) + 3

3 + 2 sec^{2} (x) - 7 sec (x) = 0

Usando el método de sustitución

3 + 2 sec^{2} (x) - 7 sec (x) = 0

Sea:

sec (x) = u

3 + 2 u^{2} - 7 u = 0

3 + 2 u^{2} - 7 u = 0 : u = 3, u = \frac{1}{2}

3 + 2 u^{2} - 7 u = 0

Escribir en la forma binómica

a x^{2} + b x + c = 0

2 u^{2} - 7 u + 3 = 0

Resolver con la fórmula general para ecuaciones de segundo grado:

2 u^{2} - 7 u + 3 = 0

Formula general para ecuaciones de segundo grado:

Para

a = 2, b = - 7, c = 3

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 2 \cdot 3}{2 \cdot 2}

u_{1, 2} = \frac{- ( - 7 ) \pm ( - 7 ) ^{2} - 4 \cdot 2 \cdot 3}{2 \cdot 2}

(- 7)^{2} - 4 \cdot 2 \cdot 3 = 5

(- 7)^{2} - 4 \cdot 2 \cdot 3

Aplicar las leyes de los exponentes:

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

n

es par

(- 7)^{2} = 7^{2}

= 7^{2} - 4 \cdot 2 \cdot 3

Multiplicar los numeros:

4 \cdot 2 \cdot 3 = 24

= 7^{2} - 24

7^{2} = 49

= 49 - 24

Restar:

49 - 24 = 25

= 25

Descomponer el número en factores primos:

25 = 5^{2}

= 5^{2}

Aplicar las leyes de los exponentes:

5^{2} = 5

= 5

u_{1, 2} = \frac{- ( - 7 ) \pm 5}{2 \cdot 2}

Separar las soluciones

u_{1} = \frac{- ( - 7 ) + 5}{2 \cdot 2}, u_{2} = \frac{- ( - 7 ) - 5}{2 \cdot 2}

u = \frac{- ( - 7 ) + 5}{2 \cdot 2} : 3

\frac{- ( - 7 ) + 5}{2 \cdot 2}

Aplicar la regla

- (- a) = a

= \frac{7 + 5}{2 \cdot 2}

Sumar:

7 + 5 = 12

= \frac{12}{2 \cdot 2}

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{12}{4}

Dividir:

\frac{12}{4} = 3

= 3

u = \frac{- ( - 7 ) - 5}{2 \cdot 2} : \frac{1}{2}

\frac{- ( - 7 ) - 5}{2 \cdot 2}

Aplicar la regla

- (- a) = a

= \frac{7 - 5}{2 \cdot 2}

Restar:

7 - 5 = 2

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

Multiplicar los numeros:

2 \cdot 2 = 4

= \frac{2}{4}

Eliminar los terminos comunes:

2

= \frac{1}{2}

Las soluciones a la ecuación de segundo grado son:

u = 3, u = \frac{1}{2}

Sustituir en la ecuación

u = sec (x)

sec (x) = 3, sec (x) = \frac{1}{2}

sec (x) = 3, sec (x) = \frac{1}{2}

sec (x) = 3 : x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

sec (x) = 3

Aplicar propiedades trigonométricas inversas

sec (x) = 3

Soluciones generales para

sec (x) = 3

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

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

sec (x) = \frac{1}{2} :

Sin solución

sec (x) = \frac{1}{2}

sec (x) \leq - 1 or sec (x) \geq 1

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

x = arcsec (3) + 2 πn, x = 2 π - arcsec (3) + 2 πn

Mostrar soluciones en forma decimal

x = 1.23095 \dots + 2 πn, x = 2 π - 1.23095 \dots + 2 πn

Gráfica

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