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

Solución

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

Restar

3

de ambos lados

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

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

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

Sumar elementos similares:

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

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

Agrupar términos semejantes

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

Restar:

- 3 - 1 = - 4

= 2 sec^{2} (x) - 4

= 2 sec^{2} (x) - 4

- 4 + 2 sec^{2} (x) = 0

Usando el método de sustitución

- 4 + 2 sec^{2} (x) = 0

Sea:

sec (x) = u

- 4 + 2 u^{2} = 0

- 4 + 2 u^{2} = 0 : u = 2, u = - 2

- 4 + 2 u^{2} = 0

Mover

4

al lado derecho

- 4 + 2 u^{2} = 0

Sumar

4

a ambos lados

- 4 + 2 u^{2} + 4 = 0 + 4

Simplificar

2 u^{2} = 4

2 u^{2} = 4

Dividir ambos lados entre

2

2 u^{2} = 4

Dividir ambos lados entre

2

\frac{2 u ^{2}}{2} = \frac{4}{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 (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

4sin^2(x)-6cos^2(x)=0

4 sin^{2} (x) - 6 cos^{2} (x) = 0

sin(60-a)=0.598

sin (6 0^{\circ} - a) = 0.598

sin (a) = 0.143

sin(x)=cos^2(x)+1

sin (x) = cos^{2} (x) + 1

sin (a) = 1.25