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Popular Trigonometría >

4tan^2(x)+sec(x)=-2tan^2(x)+sec^2(x)-3

Solución

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

Solución

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

Pasos de solución

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

Restar

- 2 tan^{2} (x) + sec^{2} (x) - 3

de ambos lados

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

Re-escribir usando identidades trigonométricas

3 + sec (x) - sec^{2} (x) + 6 tan^{2} (x)

Utilizar la identidad pitagórica:

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

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

= 3 + sec (x) - sec^{2} (x) + 6 (sec^{2} (x) - 1)

Simplificar

3 + sec (x) - sec^{2} (x) + 6 (sec^{2} (x) - 1) : 5 sec^{2} (x) + sec (x) - 3

3 + sec (x) - sec^{2} (x) + 6 (sec^{2} (x) - 1)

Expandir

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

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

Poner los parentesis utilizando:

a (b - c) = ab - a c

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

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

Multiplicar los numeros:

6 \cdot 1 = 6

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

= 3 + sec (x) - sec^{2} (x) + 6 sec^{2} (x) - 6

Simplificar

3 + sec (x) - sec^{2} (x) + 6 sec^{2} (x) - 6 : 5 sec^{2} (x) + sec (x) - 3

3 + sec (x) - sec^{2} (x) + 6 sec^{2} (x) - 6

Sumar elementos similares:

- sec^{2} (x) + 6 sec^{2} (x) = 5 sec^{2} (x)

= 3 + sec (x) + 5 sec^{2} (x) - 6

Agrupar términos semejantes

= sec (x) + 5 sec^{2} (x) + 3 - 6

Sumar/restar lo siguiente:

3 - 6 = - 3

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

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

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

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

Usando el método de sustitución

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

Sea:

sec (x) = u

- 3 + u + 5 u^{2} = 0

- 3 + u + 5 u^{2} = 0 : u = \frac{- 1 + 61}{10}, u = \frac{- 1 - 61}{10}

- 3 + u + 5 u^{2} = 0

Escribir en la forma binómica

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

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

Resolver con la fórmula general para ecuaciones de segundo grado:

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

Formula general para ecuaciones de segundo grado:

Para

a = 5, b = 1, c = - 3

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 5 ( - 3 )}{2 \cdot 5}

u_{1, 2} = \frac{- 1 \pm 1 ^{2} - 4 \cdot 5 ( - 3 )}{2 \cdot 5}

1^{2} - 4 \cdot 5 (- 3) = 61

1^{2} - 4 \cdot 5 (- 3)

Aplicar la regla

1^{a} = 1

1^{2} = 1

= 1 - 4 \cdot 5 (- 3)

Aplicar la regla

- (- a) = a

= 1 + 4 \cdot 5 \cdot 3

Multiplicar los numeros:

4 \cdot 5 \cdot 3 = 60

= 1 + 60

Sumar:

1 + 60 = 61

= 61

u_{1, 2} = \frac{- 1 \pm 61}{2 \cdot 5}

Separar las soluciones

u_{1} = \frac{- 1 + 61}{2 \cdot 5}, u_{2} = \frac{- 1 - 61}{2 \cdot 5}

u = \frac{- 1 + 61}{2 \cdot 5} : \frac{- 1 + 61}{10}

\frac{- 1 + 61}{2 \cdot 5}

Multiplicar los numeros:

2 \cdot 5 = 10

= \frac{- 1 + 61}{10}

u = \frac{- 1 - 61}{2 \cdot 5} : \frac{- 1 - 61}{10}

\frac{- 1 - 61}{2 \cdot 5}

Multiplicar los numeros:

2 \cdot 5 = 10

= \frac{- 1 - 61}{10}

Las soluciones a la ecuación de segundo grado son:

u = \frac{- 1 + 61}{10}, u = \frac{- 1 - 61}{10}

Sustituir en la ecuación

u = sec (x)

sec (x) = \frac{- 1 + 61}{10}, sec (x) = \frac{- 1 - 61}{10}

sec (x) = \frac{- 1 + 61}{10}, sec (x) = \frac{- 1 - 61}{10}

sec (x) = \frac{- 1 + 61}{10} :

Sin solución

sec (x) = \frac{- 1 + 61}{10}

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

Sin soluci \overset{o}{ˊ} n

sec (x) = \frac{- 1 - 61}{10} :

Sin solución

sec (x) = \frac{- 1 - 61}{10}

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

Sin soluci \overset{o}{ˊ} n

Combinar toda las soluciones

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

Gráfica

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