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Popular Trigonometría >

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

Solución

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

Solución

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

Grados

x = 0^{\circ} + 18 0^{\circ} n, x = 13 5^{\circ} + 36 0^{\circ} n, x = 22 5^{\circ} + 36 0^{\circ} n, x = 4 5^{\circ} + 36 0^{\circ} n, x = 31 5^{\circ} + 36 0^{\circ} n

Pasos de solución

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

Restar

3 tan (x)

de ambos lados

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

Factorizar

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

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

Factorizar el termino común

tan (x)

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

Factorizar

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

sec^{2} (x) - 2

Aplicar las leyes de los exponentes:

a = (a)^{2}

2 = (2)^{2}

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

Aplicar la siguiente regla para binomios al cuadrado:

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

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

= (sec (x) + 2) (sec (x) - 2)

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

tan (x) (sec (x) + 2) (sec (x) - 2) = 0

Resolver cada parte por separado

tan (x) = 0 or sec (x) + 2 = 0 or sec (x) - 2 = 0

tan (x) = 0 : x = πn

tan (x) = 0

Soluciones generales para

tan (x) = 0

tan (x)

tabla de valores periódicos con

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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = 0 + πn

x = 0 + πn

Resolver

x = 0 + πn : x = πn

x = 0 + πn

0 + πn = πn

x = πn

x = πn

sec (x) + 2 = 0 : x = \frac{3 π}{4} + 2 πn, x = \frac{5 π}{4} + 2 πn

sec (x) + 2 = 0

Desplace

2

a la derecha

sec (x) + 2 = 0

Restar

2

de ambos lados

sec (x) + 2 - 2 = 0 - 2

Simplificar

sec (x) = - 2

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

sec (x) - 2 = 0 : x = \frac{π}{4} + 2 πn, x = \frac{7 π}{4} + 2 πn

sec (x) - 2 = 0

Desplace

2

a la derecha

sec (x) - 2 = 0

Sumar

2

a ambos lados

sec (x) - 2 + 2 = 0 + 2

Simplificar

sec (x) = 2

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

Combinar toda las soluciones

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

Gráfica

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