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

Solución

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

Solución

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

+1

Grados

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

Pasos de solución

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

Restar

2 sec^{2} (x) - 3

de ambos lados

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

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

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

Agrupar términos semejantes

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

Sumar elementos similares:

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

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

Sumar/restar lo siguiente:

3 - 1 = 2

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

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

2 - sec^{2} (x) = 0

Usando el método de sustitución

2 - sec^{2} (x) = 0

Sea:

sec (x) = 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

Restar

2

de ambos lados

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

Simplificar

- u^{2} = - 2

- u^{2} = - 2

Dividir ambos lados entre

- 1

- u^{2} = - 2

Dividir ambos lados entre

- 1

\frac{- u ^{2}}{- 1} = \frac{- 2}{- 1}

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 (x)

sec (x) = 2, sec (x) = - 2

sec (x) = 2, sec (x) = - 2

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

sec (x) = 2

Soluciones generales para

sec (x) = 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}

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

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

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

sec (x) = - 2

Soluciones generales para

sec (x) = - 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}

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

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

Combinar toda las soluciones

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

Ejemplos populares

sinh(x)+4=4cosh(x)

sinh (x) + 4 = 4 cosh (x)

sin (18 x) = - 1

solvefor x,y+sin(xy)=1 resolver para

x, y + sin (x y) = 1

(sec^2(+1))(sec^2(-1))=tan(x)

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

2cos^2(x)-8cos(x)=2sin^2(x)-5

2 cos^{2} (x) - 8 cos (x) = 2 sin^{2} (x) - 5