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sec^2(x/3)=2

Solución

sec^{2} (\frac{x}{3}) = 2

Solución

x = \frac{3 π}{4} + 6 πn, x = \frac{21 π}{4} + 6 πn, x = \frac{9 π}{4} + 6 πn, x = \frac{15 π}{4} + 6 πn

Grados

x = 13 5^{\circ} + 108 0^{\circ} n, x = 94 5^{\circ} + 108 0^{\circ} n, x = 40 5^{\circ} + 108 0^{\circ} n, x = 67 5^{\circ} + 108 0^{\circ} n

Pasos de solución

sec^{2} (\frac{x}{3}) = 2

Usando el método de sustitución

sec^{2} (\frac{x}{3}) = 2

Sea:

sec (\frac{x}{3}) = u

u^{2} = 2

u^{2} = 2 : u = 2, u = - 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 (\frac{x}{3})

sec (\frac{x}{3}) = 2, sec (\frac{x}{3}) = - 2

sec (\frac{x}{3}) = 2, sec (\frac{x}{3}) = - 2

sec (\frac{x}{3}) = 2 : x = \frac{3 π}{4} + 6 πn, x = \frac{21 π}{4} + 6 πn

sec (\frac{x}{3}) = 2

Soluciones generales para

sec (\frac{x}{3}) = 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}

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

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

Resolver

\frac{x}{3} = \frac{π}{4} + 2 πn : x = \frac{3 π}{4} + 6 πn

\frac{x}{3} = \frac{π}{4} + 2 πn

Multiplicar ambos lados por

3

\frac{x}{3} = \frac{π}{4} + 2 πn

Multiplicar ambos lados por

3

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

Simplificar

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

Simplificar

\frac{3 x}{3} : x

\frac{3 x}{3}

Dividir:

\frac{3}{3} = 1

= x

Simplificar

3 \cdot \frac{π}{4} + 3 \cdot 2 πn : \frac{3 π}{4} + 6 πn

3 \cdot \frac{π}{4} + 3 \cdot 2 πn

Multiplicar

3 \cdot \frac{π}{4} : \frac{3 π}{4}

3 \cdot \frac{π}{4}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{π 3}{4}

= \frac{3 π}{4} + 3 \cdot 2 πn

Multiplicar los numeros:

3 \cdot 2 = 6

= \frac{3 π}{4} + 6 πn

x = \frac{3 π}{4} + 6 πn

x = \frac{3 π}{4} + 6 πn

x = \frac{3 π}{4} + 6 πn

Resolver

\frac{x}{3} = \frac{7 π}{4} + 2 πn : x = \frac{21 π}{4} + 6 πn

\frac{x}{3} = \frac{7 π}{4} + 2 πn

Multiplicar ambos lados por

3

\frac{x}{3} = \frac{7 π}{4} + 2 πn

Multiplicar ambos lados por

3

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

Simplificar

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

Simplificar

\frac{3 x}{3} : x

\frac{3 x}{3}

Dividir:

\frac{3}{3} = 1

= x

Simplificar

3 \cdot \frac{7 π}{4} + 3 \cdot 2 πn : \frac{21 π}{4} + 6 πn

3 \cdot \frac{7 π}{4} + 3 \cdot 2 πn

3 \cdot \frac{7 π}{4} = \frac{21 π}{4}

3 \cdot \frac{7 π}{4}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{7 π 3}{4}

Multiplicar los numeros:

7 \cdot 3 = 21

= \frac{21 π}{4}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiplicar los numeros:

3 \cdot 2 = 6

= 6 πn

= \frac{21 π}{4} + 6 πn

x = \frac{21 π}{4} + 6 πn

x = \frac{21 π}{4} + 6 πn

x = \frac{21 π}{4} + 6 πn

x = \frac{3 π}{4} + 6 πn, x = \frac{21 π}{4} + 6 πn

sec (\frac{x}{3}) = - 2 : x = \frac{9 π}{4} + 6 πn, x = \frac{15 π}{4} + 6 πn

sec (\frac{x}{3}) = - 2

Soluciones generales para

sec (\frac{x}{3}) = - 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}

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

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

Resolver

\frac{x}{3} = \frac{3 π}{4} + 2 πn : x = \frac{9 π}{4} + 6 πn

\frac{x}{3} = \frac{3 π}{4} + 2 πn

Multiplicar ambos lados por

3

\frac{x}{3} = \frac{3 π}{4} + 2 πn

Multiplicar ambos lados por

3

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

Simplificar

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

Simplificar

\frac{3 x}{3} : x

\frac{3 x}{3}

Dividir:

\frac{3}{3} = 1

= x

Simplificar

3 \cdot \frac{3 π}{4} + 3 \cdot 2 πn : \frac{9 π}{4} + 6 πn

3 \cdot \frac{3 π}{4} + 3 \cdot 2 πn

3 \cdot \frac{3 π}{4} = \frac{9 π}{4}

3 \cdot \frac{3 π}{4}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{3 π 3}{4}

Multiplicar los numeros:

3 \cdot 3 = 9

= \frac{9 π}{4}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiplicar los numeros:

3 \cdot 2 = 6

= 6 πn

= \frac{9 π}{4} + 6 πn

x = \frac{9 π}{4} + 6 πn

x = \frac{9 π}{4} + 6 πn

x = \frac{9 π}{4} + 6 πn

Resolver

\frac{x}{3} = \frac{5 π}{4} + 2 πn : x = \frac{15 π}{4} + 6 πn

\frac{x}{3} = \frac{5 π}{4} + 2 πn

Multiplicar ambos lados por

3

\frac{x}{3} = \frac{5 π}{4} + 2 πn

Multiplicar ambos lados por

3

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

Simplificar

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

Simplificar

\frac{3 x}{3} : x

\frac{3 x}{3}

Dividir:

\frac{3}{3} = 1

= x

Simplificar

3 \cdot \frac{5 π}{4} + 3 \cdot 2 πn : \frac{15 π}{4} + 6 πn

3 \cdot \frac{5 π}{4} + 3 \cdot 2 πn

3 \cdot \frac{5 π}{4} = \frac{15 π}{4}

3 \cdot \frac{5 π}{4}

Multiplicar fracciones:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{5 π 3}{4}

Multiplicar los numeros:

5 \cdot 3 = 15

= \frac{15 π}{4}

3 \cdot 2 πn = 6 πn

3 \cdot 2 πn

Multiplicar los numeros:

3 \cdot 2 = 6

= 6 πn

= \frac{15 π}{4} + 6 πn

x = \frac{15 π}{4} + 6 πn

x = \frac{15 π}{4} + 6 πn

x = \frac{15 π}{4} + 6 πn

x = \frac{9 π}{4} + 6 πn, x = \frac{15 π}{4} + 6 πn

Combinar toda las soluciones

x = \frac{3 π}{4} + 6 πn, x = \frac{21 π}{4} + 6 πn, x = \frac{9 π}{4} + 6 πn, x = \frac{15 π}{4} + 6 πn

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