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Popular Trigonometría >

tan^2(x)+5=3sec^2(x)

Solución

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

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

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

Restar

3 sec^{2} (x)

de ambos lados

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

Re-escribir usando identidades trigonométricas

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

Utilizar la identidad pitagórica:

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

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

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

Simplificar

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

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

Agrupar términos semejantes

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

Sumar elementos similares:

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

= - 2 sec^{2} (x) + 5 - 1

Sumar/restar lo siguiente:

5 - 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

Desplace

4

a la derecha

4 - 2 u^{2} = 0

Restar

4

de 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

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

Simplificar

\frac{- 2 u ^{2}}{- 2} : u^{2}

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

Aplicar las propiedades de las fracciones:

\frac{- a}{- b} = \frac{a}{b}

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

Dividir:

\frac{2}{2} = 1

= u^{2}

Simplificar

\frac{- 4}{- 2} : 2

\frac{- 4}{- 2}

Aplicar las propiedades de las fracciones:

\frac{- a}{- b} = \frac{a}{b}

= \frac{4}{2}

Dividir:

\frac{4}{2} = 2

= 2

u^{2} = 2

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

Gráfica

Ejemplos populares

tan(x)= 7/2

cos(θ)= 35/37

sin(θ)=-0.2761

8^2=17^2+15^2-2(17)(15)cos(A)

tan(x)=-0.35