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3sec^2(θ)-2tan^2(θ)-4=0

Solución

3 sec^{2} (θ) - 2 tan^{2} (θ) - 4 = 0

Solución

θ = \frac{π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn, θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

Grados

θ = 4 5^{\circ} + 36 0^{\circ} n, θ = 31 5^{\circ} + 36 0^{\circ} n, θ = 13 5^{\circ} + 36 0^{\circ} n, θ = 22 5^{\circ} + 36 0^{\circ} n

Pasos de solución

3 sec^{2} (θ) - 2 tan^{2} (θ) - 4 = 0

Re-escribir usando identidades trigonométricas

- 4 - 2 tan^{2} (θ) + 3 sec^{2} (θ)

Utilizar la identidad pitagórica:

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

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

= - 4 - 2 (sec^{2} (θ) - 1) + 3 sec^{2} (θ)

Simplificar

- 4 - 2 (sec^{2} (θ) - 1) + 3 sec^{2} (θ) : sec^{2} (θ) - 2

- 4 - 2 (sec^{2} (θ) - 1) + 3 sec^{2} (θ)

Expandir

- 2 (sec^{2} (θ) - 1) : - 2 sec^{2} (θ) + 2

- 2 (sec^{2} (θ) - 1)

Poner los parentesis utilizando:

a (b - c) = ab - a c

a = - 2, b = sec^{2} (θ), c = 1

= - 2 sec^{2} (θ) - (- 2) \cdot 1

Aplicar las reglas de los signos

- (- a) = a

= - 2 sec^{2} (θ) + 2 \cdot 1

Multiplicar los numeros:

2 \cdot 1 = 2

= - 2 sec^{2} (θ) + 2

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

Simplificar

- 4 - 2 sec^{2} (θ) + 2 + 3 sec^{2} (θ) : sec^{2} (θ) - 2

- 4 - 2 sec^{2} (θ) + 2 + 3 sec^{2} (θ)

Agrupar términos semejantes

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

Sumar elementos similares:

- 2 sec^{2} (θ) + 3 sec^{2} (θ) = sec^{2} (θ)

= sec^{2} (θ) - 4 + 2

Sumar/restar lo siguiente:

- 4 + 2 = - 2

= sec^{2} (θ) - 2

= sec^{2} (θ) - 2

= sec^{2} (θ) - 2

- 2 + sec^{2} (θ) = 0

Usando el método de sustitución

- 2 + sec^{2} (θ) = 0

Sea:

sec (θ) = u

- 2 + u^{2} = 0

- 2 + u^{2} = 0 : u = 2, u = - 2

- 2 + u^{2} = 0

Mover

2

al lado derecho

- 2 + u^{2} = 0

Sumar

2

a ambos lados

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

Simplificar

u^{2} = 2

u^{2} = 2

Para

x^{2} = f (a)

las soluciones son

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

u = 2, u = - 2

Sustituir en la ecuación

u = sec (θ)

sec (θ) = 2, sec (θ) = - 2

sec (θ) = 2, sec (θ) = - 2

sec (θ) = 2 : θ = \frac{π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

sec (θ) = 2

Soluciones generales para

sec (θ) = 2

sec (x)

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

θ = \frac{π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

θ = \frac{π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn

sec (θ) = - 2 : θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

sec (θ) = - 2

Soluciones generales para

sec (θ) = - 2

sec (x)

tabla de valores periódicos con

2 πn

intervalos:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sec (x) 1 \frac{2 3}{3} 22 Undefined - 2 - 2 - \frac{2 3}{3} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sec (x) - 1 - \frac{2 3}{3} - 2 - 2 Undefined 22 \frac{2 3}{3}

θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

Combinar toda las soluciones

θ = \frac{π}{4} + 2 πn, θ = \frac{7 π}{4} + 2 πn, θ = \frac{3 π}{4} + 2 πn, θ = \frac{5 π}{4} + 2 πn

Ejemplos populares

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

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

(sin(x))^{(sin(x))}= 1/(sqrt(2))

(sin (x))^{(s i n (x))} = \frac{1}{2}

(d^2-4)=cos^2(x)

(d^{2} - 4) = cos^{2} (x)

cos(3θ)=4cos(3θ)-3cos(θ)

cos (3 θ) = 4 cos (3 θ) - 3 cos (θ)

sin^2(x)-7sin(x)=0

sin^{2} (x) - 7 sin (x) = 0